Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because division by zero is undefined. So, I need to find the values of x that make the denominator zero. The denominator here is x² - 9.

Alright, so let me set the denominator equal to zero and solve for x. That would be x² - 9 = 0. Let me solve this equation.

First, I can factor x² - 9. Wait, isn't that a difference of squares? Yes! Because x² is a square and 9 is 3 squared. So, the difference of squares formula is a² - b² = (a - b)(a + b). Applying that here, x² - 9 factors into (x - 3)(x + 3). So, the equation becomes (x - 3)(x + 3) = 0.

Now, to solve this equation, I can use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, either x - 3 = 0 or x + 3 = 0.

Solving x - 3 = 0 gives x = 3. Solving x + 3 = 0 gives x = -3.

Therefore, the denominator is zero when x is 3 or -3. That means the expression (x - 5)/(x² - 9) is undefined at x = 3 and x = -3.

Wait, but the question is asking for how many values of x the expression is not defined. So, if there are two values, 3 and -3, then the answer should be 2? Let me double-check.

Let me make sure I didn't miss any other values. The denominator is a quadratic, so it can have at most two real roots. Since we factored it into (x - 3)(x + 3), which are two distinct linear factors, there are exactly two real roots. Therefore, there are exactly two values of x that make the denominator zero.

But hold on, let me check if the numerator is also zero for any of these x values. Because if the numerator is also zero, then the expression would be 0/0, which is indeterminate, but technically, the expression is still undefined. However, the question is about when the expression is not defined. So even if the numerator is zero, if the denominator is zero, the expression is undefined.

So, let me check for x = 3: numerator is 3 - 5 = -2, denominator is 0. So, it's -2/0, which is undefined. Similarly, for x = -3: numerator is -3 - 5 = -8, denominator is 0. So, -8/0, which is also undefined. Therefore, both x = 3 and x = -3 make the expression undefined.

Therefore, there are two values of x where the expression is not defined.

Wait, but let me think again. Is there a possibility that the denominator could be zero for more values? For example, if the denominator was a higher degree polynomial, but here it's quadratic. So, maximum two real roots. Since we found two real roots, that's all.

Alternatively, maybe I can solve x² - 9 = 0 by another method. Let me try solving it as x² = 9. Then, taking square roots, x = ±√9, so x = 3 or x = -3. Yep, same result.

Therefore, there are two values.

But let me check if the original problem had any restrictions or something else. The problem says "for how many values of x is the expression... not defined". So, the expression is not defined when denominator is zero, regardless of the numerator. So, even if numerator is zero, as long as denominator is zero, it's undefined.

So, even if there was a case where numerator and denominator both zero, it's still undefined. So, in this case, since numerator is -2 and -8, not zero, so no issues.

Therefore, the answer is 2.

But wait, let me check once more. Maybe I made a mistake in factoring? Let me expand (x - 3)(x + 3) to check.

(x - 3)(x + 3) = x*x + x*3 - 3*x - 3*3 = x² + 3x - 3x - 9 = x² - 9. Yes, that's correct.

So, the denominator factors correctly.

Alternatively, maybe the problem is written differently? Let me check the original problem again.

"For how many values of x is the expression (x - 5)/(x² - 9) not defined?"

Yes, that's what it says. So, denominator is x² - 9.

Therefore, the answer is 2.

But wait, hold on. Wait, sometimes people might think about the domain of the expression. The expression is not defined at x = 3 and x = -3. So, those are two points. Therefore, the number of values is 2.

Alternatively, maybe the problem is trying to trick me? For example, if there was a common factor between numerator and denominator, but in this case, numerator is x - 5, denominator is x² - 9. Let me check if they have any common factors.

Let me factor numerator and denominator. Numerator is x - 5, which is linear. Denominator is (x - 3)(x + 3). So, there are no common factors between numerator and denominator. Therefore, there are no holes in the graph, just vertical asymptotes at x = 3 and x = -3.

Therefore, the expression is undefined at exactly two points.

Therefore, the answer is 2.

But just to be thorough, let me check if there are any other points where the expression might be undefined. For example, if there was a variable in the denominator's exponent or something else, but no, the denominator is just x² - 9, which is a polynomial. Polynomials are defined for all real numbers, except where they are zero. So, the only points where the expression is undefined are where denominator is zero.

Therefore, the answer is 2.

Wait, but maybe the question is from a complex numbers perspective? But no, usually in these problems, unless specified, we assume x is real. So, in real numbers, x² - 9 = 0 has two real solutions.

Therefore, the answer is 2.

I think that's solid. I can't think of any other way this could be interpreted. So, I think the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression
$$
\frac{x - 5}{x^2 - 9}
$$
is **not defined**, we need to identify when the **denominator becomes zero**, since division by zero is undefined in mathematics.

---

### Step 1: Analyze the Denominator

The denominator of the expression is:
$$
x^2 - 9
$$

This is a **difference of squares**, which can be factored as:
$$
x^2 - 9 = (x - 3)(x + 3)
$$

---

### Step 2: Solve for When the Denominator is Zero

We set the denominator equal to zero and solve:
$$
(x - 3)(x + 3) = 0
$$

Using the **zero product property**, we find:
$$
x - 3 = 0 \quad \Rightarrow \quad x = 3 \\
x + 3 = 0 \quad \Rightarrow \quad x = -3
$$

So, the expression is **undefined** when $ x = 3 $ or $ x = -3 $.

---

### Step 3: Confirm No Additional Undefined Points

The numerator of the expression is $ x - 5 $, which is a linear expression and is defined for all real numbers. Therefore, the only points where the expression is undefined are those that make the denominator zero.

We also check whether the numerator is zero at these points:
- At $ x = 3 $: $ x - 5 = -2 $ (not zero)
- At $ x = -3 $: $ x - 5 = -8 $ (not zero)

Thus, the expression is **not defined** at both $ x = 3 $ and $ x = -3 $, and these are the **only** such points.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because division by zero is undefined. So, I need to find the values of x that make the denominator zero. The denominator here is x² - 9.

Alright, so let me set the denominator equal to zero and solve for x. That would be x² - 9 = 0. Let me solve this equation.

First, I can factor x² - 9. Wait, isn't that a difference of squares? Yes! Because x² is a square and 9 is 3 squared. So, the difference of squares formula is a² - b² = (a - b)(a + b). Applying that here, x² - 9 factors into (x - 3)(x + 3). So, the equation becomes (x - 3)(x + 3) = 0.

Now, to solve this equation, I can use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, either x - 3 = 0 or x + 3 = 0.

Solving x - 3 = 0 gives x = 3. Solving x + 3 = 0 gives x = -3.

Therefore, the denominator is zero when x is 3 or -3. That means the expression (x - 5)/(x² - 9) is undefined at x = 3 and x = -3.

Wait, but the question is asking for how many values of x the expression is not defined. So, if there are two values, 3 and -3, then the answer should be 2? Let me double-check.

Let me make sure I didn't miss any other values. The denominator is a quadratic, so it can have at most two real roots. Since we factored it into (x - 3)(x + 3), which are two distinct linear factors, there are exactly two real roots. Therefore, there are exactly two values of x that make the denominator zero.

But hold on, let me check if the numerator is also zero for any of these x values. Because if the numerator is also zero, then the expression would be 0/0, which is indeterminate, but technically, the expression is still undefined. However, the question is about when the expression is not defined. So even if the numerator is zero, if the denominator is zero, the expression is undefined.

So, let me check for x = 3: numerator is 3 - 5 = -2, denominator is 0. So, it's -2/0, which is undefined. Similarly, for x = -3: numerator is -3 - 5 = -8, denominator is 0. So, -8/0, which is also undefined. Therefore, both x = 3 and x = -3 make the expression undefined.

Therefore, there are two values of x where the expression is not defined.

Wait, but let me think again. Is there a possibility that the denominator could be zero for more values? For example, if the denominator was a higher degree polynomial, but here it's quadratic. So, maximum two real roots. Since we found two real roots, that's all.

Alternatively, maybe I can solve x² - 9 = 0 by another method. Let me try solving it as x² = 9. Then, taking square roots, x = ±√9, so x = 3 or x = -3. Yep, same result.

Therefore, there are two values.

But let me check if the original problem had any restrictions or something else. The problem says "for how many values of x is the expression... not defined". So, the expression is not defined when denominator is zero, regardless of the numerator. So, even if numerator is zero, as long as denominator is zero, it's undefined.

So, even if there was a case where numerator and denominator both zero, it's still undefined. So, in this case, since numerator is -2 and -8, not zero, so no issues.

Therefore, the answer is 2.

But wait, let me check once more. Maybe I made a mistake in factoring? Let me expand (x - 3)(x + 3) to check.

(x - 3)(x + 3) = x*x + x*3 - 3*x - 3*3 = x² + 3x - 3x - 9 = x² - 9. Yes, that's correct.

So, the denominator factors correctly.

Alternatively, maybe the problem is written differently? Let me check the original problem again.

"For how many values of x is the expression (x - 5)/(x² - 9) not defined?"

Yes, that's what it says. So, denominator is x² - 9.

Therefore, the answer is 2.

But wait, hold on. Wait, sometimes people might think about the domain of the expression. The expression is not defined at x = 3 and x = -3. So, those are two points. Therefore, the number of values is 2.

Alternatively, maybe the problem is trying to trick me? For example, if there was a common factor between numerator and denominator, but in this case, numerator is x - 5, denominator is x² - 9. Let me check if they have any common factors.

Let me factor numerator and denominator. Numerator is x - 5, which is linear. Denominator is (x - 3)(x + 3). So, there are no common factors between numerator and denominator. Therefore, there are no holes in the graph, just vertical asymptotes at x = 3 and x = -3.

Therefore, the expression is undefined at exactly two points.

Therefore, the answer is 2.

But just to be thorough, let me check if there are any other points where the expression might be undefined. For example, if there was a variable in the denominator's exponent or something else, but no, the denominator is just x² - 9, which is a polynomial. Polynomials are defined for all real numbers, except where they are zero. So, the only points where the expression is undefined are where denominator is zero.

Therefore, the answer is 2.

Wait, but maybe the question is from a complex numbers perspective? But no, usually in these problems, unless specified, we assume x is real. So, in real numbers, x² - 9 = 0 has two real solutions.

Therefore, the answer is 2.

I think that's solid. I can't think of any other way this could be interpreted. So, I think the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression
$$
\frac{x - 5}{x^2 - 9}
$$
is **not defined**, we need to identify when the **denominator becomes zero**, since division by zero is undefined in mathematics.

---

### Step 1: Analyze the Denominator

The denominator of the expression is:
$$
x^2 - 9
$$

This is a **difference of squares**, which can be factored as:
$$
x^2 - 9 = (x - 3)(x + 3)
$$

---

### Step 2: Solve for When the Denominator is Zero

We set the denominator equal to zero and solve:
$$
(x - 3)(x + 3) = 0
$$

Using the **zero product property**, we find:
$$
x - 3 = 0 \quad \Rightarrow \quad x = 3 \\
x + 3 = 0 \quad \Rightarrow \quad x = -3
$$

So, the expression is **undefined** when $ x = 3 $ or $ x = -3 $.

---

### Step 3: Confirm No Additional Undefined Points

The numerator of the expression is $ x - 5 $, which is a linear expression and is defined for all real numbers. Therefore, the only points where the expression is undefined are those that make the denominator zero.

We also check whether the numerator is zero at these points:
- At $ x = 3 $: $ x - 5 = -2 $ (not zero)
- At $ x = -3 $: $ x - 5 = -8 $ (not zero)

Thus, the expression is **not defined** at both $ x = 3 $ and $ x = -3 $, and these are the **only** such points.

---

### Final Answer

$$
\boxed{2}
$$

Solution: The expression is not defined when the denominator is equal to zero. Thus, we need to find the number of values of $x$ such that $x^2-9=0$. Rearranging the equation and taking the square root of both sides, we have $x^2-9=0\Rightarrow x^2=9 \Rightarrow x=\pm3$. Thus, there are $\boxed{2}$ different values of $x$ that make the given expression undefined.


Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize it first. The Asymptote code draws a rectangle with length 10 cm on the bottom and top, and height 3 cm on the sides. Then there's a triangle connecting the bottom-left corner (0,0) to the point (2,3) and then to the bottom-right corner (10,0). The shaded area is this triangle.

First, let me recall that the area of a triangle is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. However, since the triangle isn't necessarily aligned with the axes, maybe I need to use coordinates to calculate it?

Alternatively, maybe I can use coordinates to find the area. The coordinates of the triangle are (0,0), (2,3), and (10,0). So if I can use the coordinates, I can apply the shoelace formula or something like that. Let me try that.

The shoelace formula for three points (x1,y1), (x2,y2), (x3,y3) is:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|

Let me plug in the coordinates:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So plugging into the formula:

Area = |(0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3))/2|

Calculating each term:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 - 30 = -30

Take absolute value and divide by 2: |-30| / 2 = 30 / 2 = 15

So the area is 15 cm²? Wait, but let me check if that's correct. Let me think again.

Alternatively, maybe I can divide the figure into parts. The rectangle has area 10 cm * 3 cm = 30 cm². The shaded triangle is part of this rectangle. If I can find the area of the triangle by subtracting other areas?

But maybe not. Alternatively, since the triangle is formed by points (0,0), (2,3), (10,0). Let me try to find the base and height. If I take the base as the side from (0,0) to (10,0), which is 10 cm. Then the height would be the vertical distance from this base to the point (2,3). But wait, the base is along the x-axis, so the height would be the y-coordinate of the point (2,3), which is 3 cm. But then area would be (10 * 3)/2 = 15 cm². Wait, that's the same answer as before. But that seems too straightforward. But why is that?

Wait, but if the triangle is between (0,0), (10,0), and (2,3), then isn't that triangle not necessarily having the base as 10 cm? Because the third point is not directly above the base. Wait, but if I take the base as the base from (0,0) to (10,0), then the height is the vertical distance from that base to the opposite vertex (2,3). Since the base is horizontal, the height is just the vertical coordinate of (2,3), which is 3 cm. So then area is 1/2 * base * height = 1/2 * 10 * 3 = 15 cm². So that's the same answer. So maybe that's correct?

But wait, let me think again. If I imagine the triangle with vertices at (0,0), (10,0), and (2,3). If I draw that, the height isn't necessarily 3 cm. Wait, no. Because the height is the perpendicular distance from the base to the opposite vertex. Since the base is along the x-axis, the perpendicular distance is indeed the y-coordinate of the third point. So even though the point is not directly above the midpoint of the base, the height is still 3 cm. So that seems correct. Therefore, area is 15 cm².

But let me check with another method. Let me use coordinates again. The coordinates are (0,0), (2,3), (10,0). Let me use vectors or determinants. The area can also be calculated by the determinant formula:

Area = 1/2 | (x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) | ?

Wait, maybe I need to recall the formula. Alternatively, using vectors, the area is half the magnitude of the cross product of two sides.

Let me take vectors from (0,0) to (2,3) and from (0,0) to (10,0). The cross product of these vectors is (2)(0) - (3)(10) = 0 - 30 = -30. The magnitude is 30, so area is 1/2 * 30 = 15. Same answer. So that seems to confirm it.

Alternatively, maybe I can split the triangle into two parts? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles or something else? Let me try.

Alternatively, think of the triangle as part of the rectangle. The rectangle has area 10*3=30. The triangle is shaded. If I can find the area of the other parts and subtract?

But maybe not necessary. But let me try. The triangle divides the rectangle into three parts: the shaded triangle, and two other regions. Maybe?

Alternatively, if I can find the equations of the lines and integrate or something? Probably overkill.

Alternatively, coordinates again. Let me check with shoelace formula again. Let me write down the coordinates in order:

(0,0), (2,3), (10,0), back to (0,0). Shoelace formula:

Multiply and sum:

x1*y2 = 0*3 = 0

x2*y3 = 2*0 = 0

x3*y1 = 10*0 = 0

Sum of these: 0 + 0 + 0 = 0

Minus:

y1*x2 = 0*2 = 0

y2*x3 = 3*10 = 30

y3*x1 = 0*0 = 0

Sum of these: 0 + 30 + 0 = 30

So area is |0 - 30| / 2 = 30 / 2 = 15. Same answer. So that's consistent.

So maybe the answer is 15 cm²? But let me check again. Wait, the Asymptote code draws the rectangle with length 10 cm on the top and bottom, and 3 cm on the sides. So the rectangle is 10 cm by 3 cm. Then the triangle is connecting (0,0) to (2,3) to (10,0). So the triangle is inside the rectangle.

But maybe there's a trick here? Let me think. The problem says "the area of the shaded triangle". The Asymptote code fills the triangle (0,0)--(2,3)--(10,0)--cycle. So that triangle. So according to all calculations, 15 cm². But let me check if the coordinates are correct. The rectangle is drawn from (0,0) to (10,0) to (10,3) to (0,3) to cycle. So the rectangle is 10 units wide and 3 units tall. The triangle is connecting (0,0) to (2,3) to (10,0). So the point (2,3) is on the top side of the rectangle, 2 units from the left. So the triangle is formed by the bottom-left corner, a point on the top side, and the bottom-right corner.

Alternatively, maybe I can use coordinates to find the area. Let me think of the triangle as a polygon with coordinates (0,0), (2,3), (10,0). So using the shoelace formula again, which we did, gives 15. So maybe that's correct.

But let me think again. If I imagine the triangle, from (0,0) to (2,3) to (10,0). If I draw a line from (0,0) to (2,3), which is a line going up to the top side of the rectangle, then down to (10,0). So the area of this triangle.

Alternatively, maybe I can use coordinates to find the base and height. If I take the base as the segment from (0,0) to (10,0), which is 10 cm. Then the height is the vertical distance from this base to the point (2,3). Since the base is horizontal, the vertical distance is indeed 3 cm. Therefore, area is 1/2 * 10 * 3 = 15 cm².

But wait, isn't that assuming that the height is 3 cm? But the point (2,3) is on the top edge of the rectangle, so its vertical distance to the base is 3 cm. So that's correct. Therefore, the area is 15 cm².

But let me check with another approach. Let me find the equation of the line from (0,0) to (2,3). The slope is (3 - 0)/(2 - 0) = 3/2. So the equation is y = (3/2)x. Then the line from (2,3) to (10,0). Let me find its equation. The slope is (0 - 3)/(10 - 2) = (-3)/8. So the equation is y - 3 = (-3/8)(x - 2).

Now, if I want to find the area of the triangle, maybe I can integrate or something? Probably not necessary, but let me think. Alternatively, maybe find the intersection points or something else.

Alternatively, think of the triangle as a combination of two triangles? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles? But I don't know.

Alternatively, use coordinates again. Wait, maybe I can use the formula for the area of a triangle given three points. Which is what I did before.

Alternatively, think of vectors. The vectors from (0,0) to (2,3) is (2,3), and from (0,0) to (10,0) is (10,0). The area is half the magnitude of the cross product of these vectors. The cross product in 2D is scalar magnitude |ax*by - ay*bx|, so |2*0 - 3*10| = |0 - 30| = 30. Then area is 1/2 * 30 = 15. Same answer.

So all methods lead to 15 cm². Therefore, I think the area is 15 cm².

But let me check once more. Maybe the diagram is different? The Asymptote code draws the rectangle with labels "10 cm" on the top and bottom, and "3 cm" on the left and right. So the rectangle is 10 cm in length and 3 cm in height. The triangle is connecting (0,0) to (2,3) to (10,0). So the coordinates are correct.

Alternatively, maybe the problem is in the units? The labels are 10 cm and 3 cm, so the coordinates are in cm. So the area would be 15 cm².

Alternatively, maybe I made a mistake in assuming the base is 10 cm? Let me think. If I take the base as the segment from (0,0) to (10,0), which is 10 cm. The height is the perpendicular distance from this base to the opposite vertex (2,3). Since the base is horizontal, the perpendicular distance is vertical, so yes, 3 cm. Therefore, area is 1/2 * 10 * 3 = 15.

Alternatively, if I take another side as the base. For example, take the side from (0,0) to (2,3) as the base. Then the length of this base is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13). Then I need the height corresponding to this base. But that seems more complicated. Let me try.

The area is 15, so if I take base as sqrt(13), then height would be (2*Area)/base = 30 / sqrt(13). But that's not necessary.

Alternatively, maybe the problem is expecting me to use some other method? For example, coordinate geometry.

Alternatively, maybe the diagram is misleading? Let me check the Asymptote code again.

The Asymptote code:

fill((0,0)--(2,3)--(10,0)--cycle,gray);

draw((0,0)--(10,0)--(10,3)--(0,3)--cycle,linewidth(1));

draw((0,0)--(2,3)--(10,0),linewidth(1));

label("10 cm",(5,3),N);

label("10 cm",(5,0),S);

label("3 cm",(0,1.5),W);

label("3 cm",(10,1.5),E);

draw((0,2.5)--(.5,2.5)--(.5,3));

draw((10,2.5)--(9.5,2.5)--(9.5,3));

So the rectangle is drawn with length 10 cm on the top and bottom (from (0,0) to (10,0) and (0,3) to (10,3)), and height 3 cm on the sides. The labels "10 cm" are on the top and bottom, and "3 cm" on the left and right. The triangle is drawn from (0,0) to (2,3) to (10,0). The point (2,3) is 2 units from the left on the top side.

So the coordinates are consistent with the labels. The rectangle is 10 cm wide and 3 cm tall. The triangle is inside.

Therefore, all calculations point to 15 cm².

But let me think again. Maybe the problem is in the fact that the triangle is not entirely within the rectangle? No, the triangle is connecting (0,0) to (2,3) to (10,0), which are all points on the rectangle. So the triangle is entirely within the rectangle.

Alternatively, maybe the answer is different? Let me check with coordinates again.

Alternatively, maybe I can use coordinates to find the area by subtracting areas of other triangles. For example, the rectangle has area 30. The shaded triangle is part of it. If I can find the area of the other regions and subtract.

Let me try. The rectangle is divided by the triangle into three regions: the shaded triangle, and two other regions. Let me think.

From the rectangle, if I draw the triangle (0,0)-(2,3)-(10,0), then the rectangle is divided into the triangle and two quadrilaterals? Or maybe two triangles and a quadrilateral?

Alternatively, maybe split the rectangle into the triangle and other shapes.

Alternatively, think of the triangle as the union of two triangles: (0,0)-(2,3)-(0,3) and (0,3)-(2,3)-(10,0)? Not sure.

Alternatively, maybe find the area of the triangle by coordinates.

Alternatively, think of the triangle as a trapezoid minus something.

Alternatively, use coordinates again.

Alternatively, think of the triangle as follows: from (0,0) to (2,3) to (10,0). If I can find the area by coordinates, which we did.

Alternatively, maybe the answer is 15 cm².

But let me check with another approach. Let me use coordinates to find the area.

The coordinates are (0,0), (2,3), (10,0).

Another way to calculate area is to use the formula:

Area = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Plugging in:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So:

Area = ½ |0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3)|

= ½ |0 + 0 + 10*(-3)|

= ½ | -30 | = ½ * 30 = 15.

Same answer.

Therefore, I think the area is indeed 15 cm².

But just to be thorough, let me think if there's any chance that the triangle is not as I think. For example, maybe the coordinates are different? The Asymptote code says:

fill((0,0)--(2,3)--(10,0)--cycle,gray);

So the triangle is between (0,0), (2,3), (10,0). So that's correct.

Alternatively, maybe the labels are different? The labels say "10 cm" on the top and bottom, which are the lengths of the rectangle. The sides are labeled "3 cm", which is the height. So the rectangle is 10 cm by 3 cm.

Therefore, the coordinates are consistent with the labels.

Therefore, I think the answer is 15 cm².

But just to make sure, let me think of another way. Let me use coordinates to find the area by breaking it into parts.

Imagine the triangle (0,0), (2,3), (10,0). Let me find the area by integrating or something.

Alternatively, think of the triangle as a polygon and use the area formula.

Alternatively, use coordinates to find the base and height.

Wait, if I take the base as from (0,0) to (10,0), which is 10 cm. The height is the vertical distance from this base to the point (2,3). Since the base is along the x-axis, the height is just the y-coordinate of (2,3), which is 3 cm. Therefore, area is ½ * 10 * 3 = 15 cm².

Yes, that's straightforward.

Alternatively, if I take the base as from (0,0) to (2,3), which is sqrt( (2)^2 + (3)^2 ) = sqrt(13). Then the height would be the distance from (10,0) to the line formed by (0,0) and (2,3).

Let me calculate that. The formula for the distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, find the equation of the line through (0,0) and (2,3). The slope is 3/2, so the equation is y = (3/2)x. Rearranged: 3x - 2y = 0.

So the distance from (10,0) to this line is |3*10 - 2*0| / sqrt(3² + (-2)^2) = |30| / sqrt(13) = 30 / sqrt(13).

Then the area would be ½ * base * height = ½ * sqrt(13) * (30 / sqrt(13)) = ½ * 30 = 15.

Same answer! So regardless of which base I take, the area is 15.

Therefore, I can be confident that the area of the shaded triangle is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle in the given diagram, we are provided with the coordinates of its vertices:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

---

### Step 1: Understanding the Shape

The triangle is formed by connecting the bottom-left corner of a rectangle to a point on the top edge of the rectangle and then to the bottom-right corner. The rectangle has dimensions 10 cm (width) by 3 cm (height), and the triangle lies entirely within it.

---

### Step 2: Using the Shoelace Formula

The **shoelace formula** for the area of a triangle with vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates:

- $ x_1 = 0, y_1 = 0 $
- $ x_2 = 2, y_2 = 3 $
- $ x_3 = 10, y_3 = 0 $

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
= \frac{1}{2} \left| 0 + 0 - 30 \right|
= \frac{1}{2} \times 30 = 15
$$

---

### Step 3: Alternative Verification Using Base and Height

We can also compute the area using the standard triangle area formula:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
$$

- Take the base as the segment from $ (0, 0) $ to $ (10, 0) $, which is 10 cm.
- The height is the perpendicular distance from the point $ (2, 3) $ to the base (which lies along the x-axis), so the height is the y-coordinate of $ (2, 3) $, which is 3 cm.

$$
\text{Area} = \frac{1}{2} \times 10 \times 3 = 15
$$

---

### Step 4: Cross Product Method (Optional Verification)

Using vectors from the origin to the other two points:

- Vector $ \vec{AB} = (2, 3) $
- Vector $ \vec{AC} = (10, 0) $

The area is half the magnitude of the cross product:

$$
\text{Area} = \frac{1}{2} \left| 2 \cdot 0 - 3 \cdot 10 \right| = \frac{1}{2} \left| -30 \right| = 15
$$

---

### Final Conclusion

All methods consistently yield the same result. Therefore, the area of the shaded triangle is:

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize it first. The Asymptote code draws a rectangle with length 10 cm on the bottom and top, and height 3 cm on the sides. Then there's a triangle connecting the bottom-left corner (0,0) to the point (2,3) and then to the bottom-right corner (10,0). The shaded area is this triangle.

First, let me recall that the area of a triangle is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. However, since the triangle isn't necessarily aligned with the axes, maybe I need to use coordinates to calculate it?

Alternatively, maybe I can use coordinates to find the area. The coordinates of the triangle are (0,0), (2,3), and (10,0). So if I can use the coordinates, I can apply the shoelace formula or something like that. Let me try that.

The shoelace formula for three points (x1,y1), (x2,y2), (x3,y3) is:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|

Let me plug in the coordinates:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So plugging into the formula:

Area = |(0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3))/2|

Calculating each term:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 - 30 = -30

Take absolute value and divide by 2: |-30| / 2 = 30 / 2 = 15

So the area is 15 cm²? Wait, but let me check if that's correct. Let me think again.

Alternatively, maybe I can divide the figure into parts. The rectangle has area 10 cm * 3 cm = 30 cm². The shaded triangle is part of this rectangle. If I can find the area of the triangle by subtracting other areas?

But maybe not. Alternatively, since the triangle is formed by points (0,0), (2,3), (10,0). Let me try to find the base and height. If I take the base as the side from (0,0) to (10,0), which is 10 cm. Then the height would be the vertical distance from this base to the point (2,3). But wait, the base is along the x-axis, so the height would be the y-coordinate of the point (2,3), which is 3 cm. But then area would be (10 * 3)/2 = 15 cm². Wait, that's the same answer as before. But that seems too straightforward. But why is that?

Wait, but if the triangle is between (0,0), (10,0), and (2,3), then isn't that triangle not necessarily having the base as 10 cm? Because the third point is not directly above the base. Wait, but if I take the base as the base from (0,0) to (10,0), then the height is the vertical distance from that base to the opposite vertex (2,3). Since the base is horizontal, the height is just the vertical coordinate of (2,3), which is 3 cm. So then area is 1/2 * base * height = 1/2 * 10 * 3 = 15 cm². So that's the same answer. So maybe that's correct?

But wait, let me think again. If I imagine the triangle with vertices at (0,0), (10,0), and (2,3). If I draw that, the height isn't necessarily 3 cm. Wait, no. Because the height is the perpendicular distance from the base to the opposite vertex. Since the base is along the x-axis, the perpendicular distance is indeed the y-coordinate of the third point. So even though the point is not directly above the midpoint of the base, the height is still 3 cm. So that seems correct. Therefore, area is 15 cm².

But let me check with another method. Let me use coordinates again. The coordinates are (0,0), (2,3), (10,0). Let me use vectors or determinants. The area can also be calculated by the determinant formula:

Area = 1/2 | (x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) | ?

Wait, maybe I need to recall the formula. Alternatively, using vectors, the area is half the magnitude of the cross product of two sides.

Let me take vectors from (0,0) to (2,3) and from (0,0) to (10,0). The cross product of these vectors is (2)(0) - (3)(10) = 0 - 30 = -30. The magnitude is 30, so area is 1/2 * 30 = 15. Same answer. So that seems to confirm it.

Alternatively, maybe I can split the triangle into two parts? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles or something else? Let me try.

Alternatively, think of the triangle as part of the rectangle. The rectangle has area 10*3=30. The triangle is shaded. If I can find the area of the other parts and subtract?

But maybe not necessary. But let me try. The triangle divides the rectangle into three parts: the shaded triangle, and two other regions. Maybe?

Alternatively, if I can find the equations of the lines and integrate or something? Probably overkill.

Alternatively, coordinates again. Let me check with shoelace formula again. Let me write down the coordinates in order:

(0,0), (2,3), (10,0), back to (0,0). Shoelace formula:

Multiply and sum:

x1*y2 = 0*3 = 0

x2*y3 = 2*0 = 0

x3*y1 = 10*0 = 0

Sum of these: 0 + 0 + 0 = 0

Minus:

y1*x2 = 0*2 = 0

y2*x3 = 3*10 = 30

y3*x1 = 0*0 = 0

Sum of these: 0 + 30 + 0 = 30

So area is |0 - 30| / 2 = 30 / 2 = 15. Same answer. So that's consistent.

So maybe the answer is 15 cm²? But let me check again. Wait, the Asymptote code draws the rectangle with length 10 cm on the top and bottom, and 3 cm on the sides. So the rectangle is 10 cm by 3 cm. Then the triangle is connecting (0,0) to (2,3) to (10,0). So the triangle is inside the rectangle.

But maybe there's a trick here? Let me think. The problem says "the area of the shaded triangle". The Asymptote code fills the triangle (0,0)--(2,3)--(10,0)--cycle. So that triangle. So according to all calculations, 15 cm². But let me check if the coordinates are correct. The rectangle is drawn from (0,0) to (10,0) to (10,3) to (0,3) to cycle. So the rectangle is 10 units wide and 3 units tall. The triangle is connecting (0,0) to (2,3) to (10,0). So the point (2,3) is on the top side of the rectangle, 2 units from the left. So the triangle is formed by the bottom-left corner, a point on the top side, and the bottom-right corner.

Alternatively, maybe I can use coordinates to find the area. Let me think of the triangle as a polygon with coordinates (0,0), (2,3), (10,0). So using the shoelace formula again, which we did, gives 15. So maybe that's correct.

But let me think again. If I imagine the triangle, from (0,0) to (2,3) to (10,0). If I draw a line from (0,0) to (2,3), which is a line going up to the top side of the rectangle, then down to (10,0). So the area of this triangle.

Alternatively, maybe I can use coordinates to find the base and height. If I take the base as the segment from (0,0) to (10,0), which is 10 cm. Then the height is the vertical distance from this base to the point (2,3). Since the base is horizontal, the vertical distance is indeed 3 cm. Therefore, area is 1/2 * 10 * 3 = 15 cm².

But wait, isn't that assuming that the height is 3 cm? But the point (2,3) is on the top edge of the rectangle, so its vertical distance to the base is 3 cm. So that's correct. Therefore, the area is 15 cm².

But let me check with another approach. Let me find the equation of the line from (0,0) to (2,3). The slope is (3 - 0)/(2 - 0) = 3/2. So the equation is y = (3/2)x. Then the line from (2,3) to (10,0). Let me find its equation. The slope is (0 - 3)/(10 - 2) = (-3)/8. So the equation is y - 3 = (-3/8)(x - 2).

Now, if I want to find the area of the triangle, maybe I can integrate or something? Probably not necessary, but let me think. Alternatively, maybe find the intersection points or something else.

Alternatively, think of the triangle as a combination of two triangles? For example, from (0,0) to (2,3) to (10,0). Maybe split it into two triangles? But I don't know.

Alternatively, use coordinates again. Wait, maybe I can use the formula for the area of a triangle given three points. Which is what I did before.

Alternatively, think of vectors. The vectors from (0,0) to (2,3) is (2,3), and from (0,0) to (10,0) is (10,0). The area is half the magnitude of the cross product of these vectors. The cross product in 2D is scalar magnitude |ax*by - ay*bx|, so |2*0 - 3*10| = |0 - 30| = 30. Then area is 1/2 * 30 = 15. Same answer.

So all methods lead to 15 cm². Therefore, I think the area is 15 cm².

But let me check once more. Maybe the diagram is different? The Asymptote code draws the rectangle with labels "10 cm" on the top and bottom, and "3 cm" on the left and right. So the rectangle is 10 cm in length and 3 cm in height. The triangle is connecting (0,0) to (2,3) to (10,0). So the coordinates are correct.

Alternatively, maybe the problem is in the units? The labels are 10 cm and 3 cm, so the coordinates are in cm. So the area would be 15 cm².

Alternatively, maybe I made a mistake in assuming the base is 10 cm? Let me think. If I take the base as the segment from (0,0) to (10,0), which is 10 cm. The height is the perpendicular distance from this base to the opposite vertex (2,3). Since the base is horizontal, the perpendicular distance is vertical, so yes, 3 cm. Therefore, area is 1/2 * 10 * 3 = 15.

Alternatively, if I take another side as the base. For example, take the side from (0,0) to (2,3) as the base. Then the length of this base is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13). Then I need the height corresponding to this base. But that seems more complicated. Let me try.

The area is 15, so if I take base as sqrt(13), then height would be (2*Area)/base = 30 / sqrt(13). But that's not necessary.

Alternatively, maybe the problem is expecting me to use some other method? For example, coordinate geometry.

Alternatively, maybe the diagram is misleading? Let me check the Asymptote code again.

The Asymptote code:

fill((0,0)--(2,3)--(10,0)--cycle,gray);

draw((0,0)--(10,0)--(10,3)--(0,3)--cycle,linewidth(1));

draw((0,0)--(2,3)--(10,0),linewidth(1));

label("10 cm",(5,3),N);

label("10 cm",(5,0),S);

label("3 cm",(0,1.5),W);

label("3 cm",(10,1.5),E);

draw((0,2.5)--(.5,2.5)--(.5,3));

draw((10,2.5)--(9.5,2.5)--(9.5,3));

So the rectangle is drawn with length 10 cm on the top and bottom (from (0,0) to (10,0) and (0,3) to (10,3)), and height 3 cm on the sides. The labels "10 cm" are on the top and bottom, and "3 cm" on the left and right. The triangle is drawn from (0,0) to (2,3) to (10,0). The point (2,3) is 2 units from the left on the top side.

So the coordinates are consistent with the labels. The rectangle is 10 cm wide and 3 cm tall. The triangle is inside.

Therefore, all calculations point to 15 cm².

But let me think again. Maybe the problem is in the fact that the triangle is not entirely within the rectangle? No, the triangle is connecting (0,0) to (2,3) to (10,0), which are all points on the rectangle. So the triangle is entirely within the rectangle.

Alternatively, maybe the answer is different? Let me check with coordinates again.

Alternatively, maybe I can use coordinates to find the area by subtracting areas of other triangles. For example, the rectangle has area 30. The shaded triangle is part of it. If I can find the area of the other regions and subtract.

Let me try. The rectangle is divided by the triangle into three regions: the shaded triangle, and two other regions. Let me think.

From the rectangle, if I draw the triangle (0,0)-(2,3)-(10,0), then the rectangle is divided into the triangle and two quadrilaterals? Or maybe two triangles and a quadrilateral?

Alternatively, maybe split the rectangle into the triangle and other shapes.

Alternatively, think of the triangle as the union of two triangles: (0,0)-(2,3)-(0,3) and (0,3)-(2,3)-(10,0)? Not sure.

Alternatively, maybe find the area of the triangle by coordinates.

Alternatively, think of the triangle as a trapezoid minus something.

Alternatively, use coordinates again.

Alternatively, think of the triangle as follows: from (0,0) to (2,3) to (10,0). If I can find the area by coordinates, which we did.

Alternatively, maybe the answer is 15 cm².

But let me check with another approach. Let me use coordinates to find the area.

The coordinates are (0,0), (2,3), (10,0).

Another way to calculate area is to use the formula:

Area = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Plugging in:

x1 = 0, y1 = 0

x2 = 2, y2 = 3

x3 = 10, y3 = 0

So:

Area = ½ |0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3)|

= ½ |0 + 0 + 10*(-3)|

= ½ | -30 | = ½ * 30 = 15.

Same answer.

Therefore, I think the area is indeed 15 cm².

But just to be thorough, let me think if there's any chance that the triangle is not as I think. For example, maybe the coordinates are different? The Asymptote code says:

fill((0,0)--(2,3)--(10,0)--cycle,gray);

So the triangle is between (0,0), (2,3), (10,0). So that's correct.

Alternatively, maybe the labels are different? The labels say "10 cm" on the top and bottom, which are the lengths of the rectangle. The sides are labeled "3 cm", which is the height. So the rectangle is 10 cm by 3 cm.

Therefore, the coordinates are consistent with the labels.

Therefore, I think the answer is 15 cm².

But just to make sure, let me think of another way. Let me use coordinates to find the area by breaking it into parts.

Imagine the triangle (0,0), (2,3), (10,0). Let me find the area by integrating or something.

Alternatively, think of the triangle as a polygon and use the area formula.

Alternatively, use coordinates to find the base and height.

Wait, if I take the base as from (0,0) to (10,0), which is 10 cm. The height is the vertical distance from this base to the point (2,3). Since the base is along the x-axis, the height is just the y-coordinate of (2,3), which is 3 cm. Therefore, area is ½ * 10 * 3 = 15 cm².

Yes, that's straightforward.

Alternatively, if I take the base as from (0,0) to (2,3), which is sqrt( (2)^2 + (3)^2 ) = sqrt(13). Then the height would be the distance from (10,0) to the line formed by (0,0) and (2,3).

Let me calculate that. The formula for the distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, find the equation of the line through (0,0) and (2,3). The slope is 3/2, so the equation is y = (3/2)x. Rearranged: 3x - 2y = 0.

So the distance from (10,0) to this line is |3*10 - 2*0| / sqrt(3² + (-2)^2) = |30| / sqrt(13) = 30 / sqrt(13).

Then the area would be ½ * base * height = ½ * sqrt(13) * (30 / sqrt(13)) = ½ * 30 = 15.

Same answer! So regardless of which base I take, the area is 15.

Therefore, I can be confident that the area of the shaded triangle is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle in the given diagram, we are provided with the coordinates of its vertices:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

---

### Step 1: Understanding the Shape

The triangle is formed by connecting the bottom-left corner of a rectangle to a point on the top edge of the rectangle and then to the bottom-right corner. The rectangle has dimensions 10 cm (width) by 3 cm (height), and the triangle lies entirely within it.

---

### Step 2: Using the Shoelace Formula

The **shoelace formula** for the area of a triangle with vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates:

- $ x_1 = 0, y_1 = 0 $
- $ x_2 = 2, y_2 = 3 $
- $ x_3 = 10, y_3 = 0 $

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
= \frac{1}{2} \left| 0 + 0 - 30 \right|
= \frac{1}{2} \times 30 = 15
$$

---

### Step 3: Alternative Verification Using Base and Height

We can also compute the area using the standard triangle area formula:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
$$

- Take the base as the segment from $ (0, 0) $ to $ (10, 0) $, which is 10 cm.
- The height is the perpendicular distance from the point $ (2, 3) $ to the base (which lies along the x-axis), so the height is the y-coordinate of $ (2, 3) $, which is 3 cm.

$$
\text{Area} = \frac{1}{2} \times 10 \times 3 = 15
$$

---

### Step 4: Cross Product Method (Optional Verification)

Using vectors from the origin to the other two points:

- Vector $ \vec{AB} = (2, 3) $
- Vector $ \vec{AC} = (10, 0) $

The area is half the magnitude of the cross product:

$$
\text{Area} = \frac{1}{2} \left| 2 \cdot 0 - 3 \cdot 10 \right| = \frac{1}{2} \left| -30 \right| = 15
$$

---

### Final Conclusion

All methods consistently yield the same result. Therefore, the area of the shaded triangle is:

$$
\boxed{15}
$$

Solution: The shaded triangle has a base of length $10\text{ cm}.$ Since the triangle is enclosed in a rectangle of height $3\text{ cm},$ then the height of the triangle is $3\text{ cm}.$ (We know that the enclosing shape is a rectangle, because any figure with four sides, including two pairs of equal opposite sides, and two right angles must be a rectangle.) Therefore, the area of the triangle is $$\frac{1}{2}\times 3 \times 10 = \boxed{15\mbox{ cm}^2}.$$


Explanation:
Okay, so Jim and Martha are at the corner of a rectangular field. The field is 300 feet wide and 400 feet long. Jim walks diagonally across the field, while Martha walks along the length and width to get to the same location. The question is asking how many feet less Jim walks compared to Martha.

First, let me visualize the problem. There's a rectangle, right? Let me sketch it mentally. Let's say the rectangle has a width of 300 feet and a length of 400 feet. So, if Jim and Martha are at one corner, let's say the bottom-left corner, then Martha would walk along the length (400 feet) and then along the width (300 feet) to reach the opposite corner. Wait, but actually, if Martha is walking along the length and width, does that mean she walks along both the length and the width? So, if the field is 400 feet long and 300 feet wide, then Martha would walk 400 feet along the length and then 300 feet along the width? But that would make her total distance 400 + 300 = 700 feet.

But Jim is taking the diagonal path. So, the distance Jim walks is the diagonal of the rectangle. To find that, I can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the rectangle's diagonal forms a right-angled triangle with the length and width as the other two sides.

So, the formula for the diagonal (let's call it d) would be:

d = √(length² + width²)

Plugging in the numbers:

d = √(400² + 300²)

Let me calculate that. First, 400 squared is 160,000, and 300 squared is 90,000. Adding those together gives 160,000 + 90,000 = 250,000. Then the square root of 250,000 is... Hmm, sqrt(250,000). Well, sqrt(250,000) is sqrt(25 * 10,000) which is sqrt(25) * sqrt(10,000) = 5 * 100 = 500. Wait, that seems straightforward. So the diagonal is 500 feet.

So Martha walks 400 + 300 = 700 feet, and Jim walks 500 feet. Therefore, Jim walks 700 - 500 = 200 feet less than Martha.

Wait, that seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, confirming the Pythagorean theorem. If the sides are 300 and 400, then the hypotenuse squared is 300² + 400². Let me compute 300 squared again: 300*300 is 90,000. 400 squared is 160,000. Adding them gives 250,000. Square root of 250,000. Let me think, 500 squared is 250,000 because 500*500 is 250,000. Yes, that's correct. So the diagonal is indeed 500 feet.

Martha's path is along the length and width. Since the field is 400 feet long and 300 feet wide, if she walks along the length and then the width, that's 400 + 300 = 700 feet. So the difference is 700 - 500 = 200 feet.

Therefore, Jim walks 200 feet less than Martha.

But let me just make sure that Martha isn't walking along both the length and the width in some other way. For example, if the field is 300 feet wide and 400 feet long, then depending on which corner she's starting from, but since they are both starting at the same corner and going to the same location, which is the opposite corner. So Martha's path is along the two sides, so yes, 400 + 300.

Alternatively, if Martha was walking along the length and then the width, but maybe she could take a different path? But since it's a rectangle, the shortest path along the sides would be the sum of the length and width. There's no shorter path along the sides, because if you go along the length and then the width, that's the only way to get from one corner to the opposite corner without cutting across.

So, yes, Martha's distance is 700 feet, Jim's is 500 feet, so the difference is 200 feet.

Wait, but let me check if the problem says "how many feet less than Martha does Jim walk?" So, Jim walks less than Martha by 200 feet. So the answer is 200.

But let me check once more. Maybe I misread the problem? Let me read again:

"Jim and Martha are standing together at the corner of a rectangular field. Jim walks diagonally across the field. Martha gets to the same location by walking along its length and width. The field is 300 feet wide and 400 feet long. How many feet less than Martha does Jim walk?"

Yes, so Martha walks along the length and width. So, if the field is 300 feet wide and 400 feet long, then Martha walks 400 feet along the length and 300 feet along the width. So total distance is 700. Jim walks the diagonal, which is 500. So 700 - 500 = 200.

Alternatively, maybe Martha walks along the width and then the length? But that's the same thing. 300 + 400 is still 700.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide and 400 feet long, then depending on the direction. But regardless, the total distance is the same.

Alternatively, could there be a different interpretation? For example, maybe Martha walks along the length and the width, but not necessarily both? But the problem says "walking along its length and width", which probably means she walks along both the length and the width.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide and 400 feet long, then if she starts at one corner, to get to the opposite corner, she would have to walk along one length and one width.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide, then maybe she walks 300 feet along the width and then 400 feet along the length? Either way, same total.

So, I think my calculation is correct.

But just to make sure, let me think of another way. Suppose the field is a rectangle with length L and width W. Then Martha walks L + W, Jim walks sqrt(L² + W²). The difference is L + W - sqrt(L² + W²).

In this case, L is 400, W is 300. So difference is 700 - 500 = 200.

Alternatively, if I plug in numbers, maybe I can check with another example. For example, if the field was a square, say 100x100. Then Martha would walk 200 feet, Jim would walk sqrt(100² + 100²) = sqrt(20000) ≈ 141.42 feet. The difference would be about 58.58 feet. Which makes sense.

Alternatively, if the field was 3-4-5 triangle, like 3-4-5 triangle. So if the field is 3 and 4, then Martha walks 7, Jim walks 5, difference is 2. Which is the same as 3+4 - 5 = 2. So that's correct.

So in our problem, since it's 300 and 400, which is a multiple of 3-4-5 triangle (times 100), so the diagonal is 500, Martha walks 700, difference is 200.

Therefore, I think the answer is 200.

But just to make sure, let me check the calculation again.

Compute 400 squared: 400*400. 4*4 is 16, and then three zeros, so 160,000.

300 squared: 3*3 is 9, three zeros, so 90,000.

Adding them: 160,000 + 90,000 = 250,000.

Square root of 250,000. Well, sqrt(250,000) = sqrt(25 * 10,000) = 5 * 100 = 500.

Yes, that's correct.

So Martha walks 400 + 300 = 700.

Difference is 700 - 500 = 200.

Therefore, Jim walks 200 feet less than Martha.

I think that's solid.

**Final Answer**
Jim walks \boxed{200} feet less than Martha.

To determine how many feet less Jim walks compared to Martha, we analyze the two different paths they take across a rectangular field.

---

### **Understanding the Problem**

- The field is a **rectangle** with:
  - **Width** = 300 feet
  - **Length** = 400 feet

- **Jim** walks **diagonally** across the field from one corner to the opposite corner.
- **Martha** walks **along the length and width** to reach the same destination.

We are asked to find how many **feet less** Jim walks than Martha.

---

### **Step 1: Calculate Martha's Distance**

Martha walks along the **length** and the **width** of the field:

$$
\text{Martha's distance} = \text{Length} + \text{Width} = 400 + 300 = 700 \text{ feet}
$$

---

### **Step 2: Calculate Jim's Distance**

Jim walks the **diagonal** of the rectangle. This forms a right triangle with the length and width as the two legs. Using the **Pythagorean Theorem**:

$$
\text{Diagonal} = \sqrt{\text{Length}^2 + \text{Width}^2}
$$

$$
\text{Diagonal} = \sqrt{400^2 + 300^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500 \text{ feet}
$$

---

### **Step 3: Find the Difference**

Now, subtract Jim's distance from Martha's distance:

$$
\text{Difference} = 700 - 500 = 200 \text{ feet}
$$

---

### **Final Answer**

$$
\boxed{200}
$$

Jim walks **200 feet less** than Martha.

[Assistant]
Explanation:
Okay, so Jim and Martha are at the corner of a rectangular field. The field is 300 feet wide and 400 feet long. Jim walks diagonally across the field, while Martha walks along the length and width to get to the same location. The question is asking how many feet less Jim walks compared to Martha.

First, let me visualize the problem. There's a rectangle, right? Let me sketch it mentally. Let's say the rectangle has a width of 300 feet and a length of 400 feet. So, if Jim and Martha are at one corner, let's say the bottom-left corner, then Martha would walk along the length (400 feet) and then along the width (300 feet) to reach the opposite corner. Wait, but actually, if Martha is walking along the length and width, does that mean she walks along both the length and the width? So, if the field is 400 feet long and 300 feet wide, then Martha would walk 400 feet along the length and then 300 feet along the width? But that would make her total distance 400 + 300 = 700 feet.

But Jim is taking the diagonal path. So, the distance Jim walks is the diagonal of the rectangle. To find that, I can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the rectangle's diagonal forms a right-angled triangle with the length and width as the other two sides.

So, the formula for the diagonal (let's call it d) would be:

d = √(length² + width²)

Plugging in the numbers:

d = √(400² + 300²)

Let me calculate that. First, 400 squared is 160,000, and 300 squared is 90,000. Adding those together gives 160,000 + 90,000 = 250,000. Then the square root of 250,000 is... Hmm, sqrt(250,000). Well, sqrt(250,000) is sqrt(25 * 10,000) which is sqrt(25) * sqrt(10,000) = 5 * 100 = 500. Wait, that seems straightforward. So the diagonal is 500 feet.

So Martha walks 400 + 300 = 700 feet, and Jim walks 500 feet. Therefore, Jim walks 700 - 500 = 200 feet less than Martha.

Wait, that seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, confirming the Pythagorean theorem. If the sides are 300 and 400, then the hypotenuse squared is 300² + 400². Let me compute 300 squared again: 300*300 is 90,000. 400 squared is 160,000. Adding them gives 250,000. Square root of 250,000. Let me think, 500 squared is 250,000 because 500*500 is 250,000. Yes, that's correct. So the diagonal is indeed 500 feet.

Martha's path is along the length and width. Since the field is 400 feet long and 300 feet wide, if she walks along the length and then the width, that's 400 + 300 = 700 feet. So the difference is 700 - 500 = 200 feet.

Therefore, Jim walks 200 feet less than Martha.

But let me just make sure that Martha isn't walking along both the length and the width in some other way. For example, if the field is 300 feet wide and 400 feet long, then depending on which corner she's starting from, but since they are both starting at the same corner and going to the same location, which is the opposite corner. So Martha's path is along the two sides, so yes, 400 + 300.

Alternatively, if Martha was walking along the length and then the width, but maybe she could take a different path? But since it's a rectangle, the shortest path along the sides would be the sum of the length and width. There's no shorter path along the sides, because if you go along the length and then the width, that's the only way to get from one corner to the opposite corner without cutting across.

So, yes, Martha's distance is 700 feet, Jim's is 500 feet, so the difference is 200 feet.

Wait, but let me check if the problem says "how many feet less than Martha does Jim walk?" So, Jim walks less than Martha by 200 feet. So the answer is 200.

But let me check once more. Maybe I misread the problem? Let me read again:

"Jim and Martha are standing together at the corner of a rectangular field. Jim walks diagonally across the field. Martha gets to the same location by walking along its length and width. The field is 300 feet wide and 400 feet long. How many feet less than Martha does Jim walk?"

Yes, so Martha walks along the length and width. So, if the field is 300 feet wide and 400 feet long, then Martha walks 400 feet along the length and 300 feet along the width. So total distance is 700. Jim walks the diagonal, which is 500. So 700 - 500 = 200.

Alternatively, maybe Martha walks along the width and then the length? But that's the same thing. 300 + 400 is still 700.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide and 400 feet long, then depending on the direction. But regardless, the total distance is the same.

Alternatively, could there be a different interpretation? For example, maybe Martha walks along the length and the width, but not necessarily both? But the problem says "walking along its length and width", which probably means she walks along both the length and the width.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide and 400 feet long, then if she starts at one corner, to get to the opposite corner, she would have to walk along one length and one width.

Alternatively, maybe Martha walks along the length and then the width, but if the field is 300 feet wide, then maybe she walks 300 feet along the width and then 400 feet along the length? Either way, same total.

So, I think my calculation is correct.

But just to make sure, let me think of another way. Suppose the field is a rectangle with length L and width W. Then Martha walks L + W, Jim walks sqrt(L² + W²). The difference is L + W - sqrt(L² + W²).

In this case, L is 400, W is 300. So difference is 700 - 500 = 200.

Alternatively, if I plug in numbers, maybe I can check with another example. For example, if the field was a square, say 100x100. Then Martha would walk 200 feet, Jim would walk sqrt(100² + 100²) = sqrt(20000) ≈ 141.42 feet. The difference would be about 58.58 feet. Which makes sense.

Alternatively, if the field was 3-4-5 triangle, like 3-4-5 triangle. So if the field is 3 and 4, then Martha walks 7, Jim walks 5, difference is 2. Which is the same as 3+4 - 5 = 2. So that's correct.

So in our problem, since it's 300 and 400, which is a multiple of 3-4-5 triangle (times 100), so the diagonal is 500, Martha walks 700, difference is 200.

Therefore, I think the answer is 200.

But just to make sure, let me check the calculation again.

Compute 400 squared: 400*400. 4*4 is 16, and then three zeros, so 160,000.

300 squared: 3*3 is 9, three zeros, so 90,000.

Adding them: 160,000 + 90,000 = 250,000.

Square root of 250,000. Well, sqrt(250,000) = sqrt(25 * 10,000) = 5 * 100 = 500.

Yes, that's correct.

So Martha walks 400 + 300 = 700.

Difference is 700 - 500 = 200.

Therefore, Jim walks 200 feet less than Martha.

I think that's solid.

**Final Answer**
Jim walks \boxed{200} feet less than Martha.

To determine how many feet less Jim walks compared to Martha, we analyze the two different paths they take across a rectangular field.

---

### **Understanding the Problem**

- The field is a **rectangle** with:
  - **Width** = 300 feet
  - **Length** = 400 feet

- **Jim** walks **diagonally** across the field from one corner to the opposite corner.
- **Martha** walks **along the length and width** to reach the same destination.

We are asked to find how many **feet less** Jim walks than Martha.

---

### **Step 1: Calculate Martha's Distance**

Martha walks along the **length** and the **width** of the field:

$$
\text{Martha's distance} = \text{Length} + \text{Width} = 400 + 300 = 700 \text{ feet}
$$

---

### **Step 2: Calculate Jim's Distance**

Jim walks the **diagonal** of the rectangle. This forms a right triangle with the length and width as the two legs. Using the **Pythagorean Theorem**:

$$
\text{Diagonal} = \sqrt{\text{Length}^2 + \text{Width}^2}
$$

$$
\text{Diagonal} = \sqrt{400^2 + 300^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500 \text{ feet}
$$

---

### **Step 3: Find the Difference**

Now, subtract Jim's distance from Martha's distance:

$$
\text{Difference} = 700 - 500 = 200 \text{ feet}
$$

---

### **Final Answer**

$$
\boxed{200}
$$

Jim walks **200 feet less** than Martha.

Solution: The diagonal of the field is $\sqrt{300^2+400^2}=500$ feet long so Jim walks 500 feet. Two adjacent sides of the field are $300+400=700$ feet long so Martha walks 700 feet. Jim walks $700-500=\boxed{200}$ feet less than Martha.


Explanation:
Okay, so I need to find all possible values of the expression $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$ where $\omega$ is a complex number such that $\omega^3 = 1$. Hmm, let me start by recalling what the roots of the equation $\omega^3 = 1$ are.

First, I know that the solutions to $\omega^3 = 1$ are the cube roots of unity. There's one real root and two complex roots. The real root is obviously 1 because $1^3 = 1$. The other two roots are complex numbers, often denoted as $\omega$ and $\omega^2$, which are complex conjugates of each other.

So, the three cube roots of unity are 1, $\omega$, and $\omega^2$, where $\omega = e^{2\pi i /3}$ and $\omega^2 = e^{4\pi i /3}$. These satisfy the equation $1 + \omega + \omega^2 = 0$. That's a key identity I remember. Maybe that will come in handy later.

Now, the problem is asking for the value of $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me note that $\omega$ can be either of the complex roots or the real root 1. Wait, but if $\omega$ is 1, then the expression becomes $\frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1$. But the problem says "all possible values," so maybe there are different values depending on which root $\omega$ is? Wait, but the problem says "Let $\omega$ be a complex number such that $\omega^3 = 1$." So $\omega$ can be any of the three roots. However, if $\omega$ is 1, then the expression is 1. If $\omega$ is a complex root, then maybe the expression evaluates to something else?

Wait, but maybe the problem is considering $\omega$ as a non-real root? Because if $\omega$ is 1, then the expression is 1, but maybe the problem wants the values when $\omega$ is not 1? Wait, the problem says "Let $\omega$ be a complex number such that $\omega^3 = 1$", so $\omega$ can be any of the three roots. Therefore, we need to check all three possibilities. However, maybe when $\omega$ is 1, the expression is 1, and when $\omega$ is the other roots, maybe the expression is the same? Or different?

Wait, let me check. Let me first consider $\omega$ as the primitive root, so $\omega \neq 1$. Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Maybe I can combine these fractions. Let me try that.

First, find a common denominator. The denominators are $1 + \omega$ and $1 + \omega^2$. The common denominator would be $(1 + \omega)(1 + \omega^2)$. Let me compute the numerator:

$1 \cdot (1 + \omega^2) + 1 \cdot (1 + \omega) = (1 + \omega^2 + 1 + \omega) = 2 + \omega + \omega^2$.

So the combined fraction is $\frac{2 + \omega + \omega^2}{(1 + \omega)(1 + \omega^2)}$.

But from the identity $1 + \omega + \omega^2 = 0$, we know that $\omega + \omega^2 = -1$. Therefore, substituting that into the numerator:

$2 + (-1) = 1$. So the numerator simplifies to 1.

Now, let me compute the denominator: $(1 + \omega)(1 + \omega^2)$. Let me expand this:

$1 \cdot 1 + 1 \cdot \omega^2 + \omega \cdot 1 + \omega \cdot \omega^2 = 1 + \omega^2 + \omega + \omega^3$.

But $\omega^3 = 1$, so substituting that in:

$1 + \omega^2 + \omega + 1 = 2 + \omega + \omega^2$. Again, using the identity $\omega + \omega^2 = -1$, this becomes $2 + (-1) = 1$.

Therefore, the denominator is 1. Therefore, the entire expression simplifies to $\frac{1}{1} = 1$. Wait, so regardless of which root $\omega$ is (as long as it's not 1?), the expression is 1? But when $\omega = 1$, we saw that the expression is also 1. So does that mean that regardless of which cube root of unity we take, the value of the expression is always 1? Therefore, the only possible value is 1?

But that seems strange. Let me check with specific values. Let me take $\omega = e^{2\pi i /3}$. Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$.

First, compute $1 + \omega$. Since $\omega = e^{2\pi i /3} = -1/2 + i\sqrt{3}/2$, so $1 + \omega = 1 - 1/2 + i\sqrt{3}/2 = 1/2 + i\sqrt{3}/2$. Similarly, $\omega^2 = e^{4\pi i /3} = -1/2 - i\sqrt{3}/2$, so $1 + \omega^2 = 1 - 1/2 - i\sqrt{3}/2 = 1/2 - i\sqrt{3}/2$.

Therefore, the expression becomes $\frac{1}{1/2 + i\sqrt{3}/2} + \frac{1}{1/2 - i\sqrt{3}/2}$. Let me compute each term.

First term: $\frac{1}{1/2 + i\sqrt{3}/2}$. Multiply numerator and denominator by the conjugate of the denominator, which is $1/2 - i\sqrt{3}/2$:

$$
\frac{1 \cdot (1/2 - i\sqrt{3}/2)}{(1/2 + i\sqrt{3}/2)(1/2 - i\sqrt{3}/2)} = \frac{1/2 - i\sqrt{3}/2}{(1/2)^2 + (\sqrt{3}/2)^2} = \frac{1/2 - i\sqrt{3}/2}{1/4 + 3/4} = \frac{1/2 - i\sqrt{3}/2}{1} = 1/2 - i\sqrt{3}/2
$$

Similarly, the second term is $\frac{1}{1/2 - i\sqrt{3}/2}$. Multiply numerator and denominator by the conjugate $1/2 + i\sqrt{3}/2$:

$$
\frac{1 \cdot (1/2 + i\sqrt{3}/2)}{(1/2 - i\sqrt{3}/2)(1/2 + i\sqrt{3}/2)} = \frac{1/2 + i\sqrt{3}/2}{1} = 1/2 + i\sqrt{3}/2
$$

Adding the two terms together: $(1/2 - i\sqrt{3}/2) + (1/2 + i\sqrt{3}/2) = 1/2 + 1/2 + (-i\sqrt{3}/2 + i\sqrt{3}/2) = 1 + 0 = 1$.

So that checks out. So even when $\omega$ is a complex root, the expression is 1. And when $\omega = 1$, as we saw earlier, the expression is also 1. Therefore, regardless of which cube root of unity we take, the value of the expression is always 1. Therefore, the only possible value is 1.

Wait, but the problem says "Find all possible values," so maybe there's only one value? But maybe I made a mistake here? Let me check again.

Alternatively, maybe there's a different approach. Let me think. Since $\omega^3 = 1$, and $\omega \neq 1$, then $\omega$ satisfies $\omega^2 + \omega + 1 = 0$. So maybe I can use that to simplify the expression.

Let me try another approach. Let me denote the expression as $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me compute $E$:

First, note that $\omega^2 = \omega^{-1}$ because $\omega^3 = 1$, so $\omega^2 = \omega^{-1}$. Similarly, $1 + \omega^2 = 1 + \omega^{-1}$.

Alternatively, maybe I can write $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me combine them:

As before, common denominator is $(1 + \omega)(1 + \omega^2)$. The numerator is $(1 + \omega^2) + (1 + \omega) = 2 + \omega + \omega^2 = 2 - 1 = 1$, since $\omega + \omega^2 = -1$. The denominator is $(1 + \omega)(1 + \omega^2)$. Let me compute that again:

Expanding $(1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega^3 = 1 + (\omega + \omega^2) + \omega^3 = 1 + (-1) + 1 = 1$. Therefore, the denominator is 1, so $E = 1/1 = 1$.

Therefore, regardless of which root $\omega$ is (even if $\omega = 1$), the expression is 1. Wait, but if $\omega = 1$, then the denominator in the original expression would be $1 + 1 = 2$ for both terms, so $1/2 + 1/2 = 1$. So even in that case, it's 1. Therefore, regardless of which cube root of unity we take, the value is always 1. Therefore, the only possible value is 1.

But the problem says "Find all possible values," so maybe there's only one value? But maybe I need to check if there are different values? Wait, but according to all calculations, it's always 1.

Alternatively, maybe there's a mistake in assuming that $\omega$ can be 1? Let me check the problem statement again. It says "Let $\omega$ be a complex number such that $\omega^3 = 1$." So $\omega$ can be 1, or the other two roots. But in all cases, the expression evaluates to 1. Therefore, the only possible value is 1.

But maybe the problem is expecting different values? Wait, let me think again. Suppose $\omega$ is a primitive cube root of unity, so $\omega \neq 1$. Then, the expression is 1. If $\omega = 1$, then the expression is also 1. Therefore, regardless of which root we take, the value is 1. Therefore, the only possible value is 1.

Alternatively, maybe there's a different interpretation? For example, maybe the problem is considering $\omega$ as a non-real root? But even then, the expression still evaluates to 1.

Alternatively, maybe I made a mistake in the algebra? Let me check again.

Let me take $\omega = 1$. Then, $\frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1$.

If $\omega$ is a non-real root, then as shown earlier, the expression is 1. Therefore, regardless of which root we take, the value is 1. Therefore, the only possible value is 1.

But maybe the problem is expecting multiple values? Wait, but according to the math, it's always 1. Let me check with another approach. Let me consider that $\omega^3 = 1$, so $\omega^3 - 1 = 0$, which factors as $(\omega - 1)(\omega^2 + \omega + 1) = 0$. Therefore, either $\omega = 1$ or $\omega^2 + \omega + 1 = 0$.

So, if $\omega \neq 1$, then $\omega^2 + \omega + 1 = 0$. Therefore, $\omega^2 = -1 - \omega$. Let me substitute this into the expression.

Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Since $\omega^2 = -1 - \omega$, then $1 + \omega^2 = 1 + (-1 - \omega) = -\omega$. Therefore, the second term becomes $\frac{1}{1 + \omega^2} = \frac{1}{-\omega} = -\frac{1}{\omega}$.

Therefore, the entire expression becomes $\frac{1}{1 + \omega} - \frac{1}{\omega}$. Let me compute this.

First, let me write $\frac{1}{1 + \omega} - \frac{1}{\omega}$. Let me combine these terms:

$$
\frac{\omega - (1 + \omega)}{\omega(1 + \omega)} = \frac{\omega - 1 - \omega}{\omega(1 + \omega)} = \frac{-1}{\omega(1 + \omega)}
$$

Hmm, so that's $\frac{-1}{\omega(1 + \omega)}$. Let me compute the denominator: $\omega(1 + \omega) = \omega + \omega^2$. But since $\omega^2 = -1 - \omega$, then $\omega + \omega^2 = \omega + (-1 - \omega) = -1$. Therefore, the denominator is $-1$, so the entire expression becomes $\frac{-1}{-1} = 1$.

So again, we get 1. Therefore, regardless of whether $\omega$ is 1 or a non-real root, the expression is 1. Therefore, the only possible value is 1.

Therefore, the answer is just 1.

But the problem says "Find all possible values," so maybe there's only one value? Maybe the problem is designed to show that even though there are three roots, the expression evaluates to the same value.

Alternatively, maybe I need to check if there's a case where the expression is undefined? For example, if $1 + \omega = 0$ or $1 + \omega^2 = 0$. But if $1 + \omega = 0$, then $\omega = -1$. But $\omega^3 = (-1)^3 = -1 \neq 1$, so $\omega = -1$ is not a root of $\omega^3 = 1$. Therefore, the denominators are never zero. Therefore, the expression is defined for all cube roots of unity.

Therefore, the conclusion is that the expression always equals 1. Therefore, the only possible value is 1.

But just to be thorough, let me check with another method. Let me consider that $\omega$ is a root of $x^3 - 1 = 0$, so $x^3 = 1$. Let me consider the expression $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me compute $E$ for each root:

Case 1: $\omega = 1$. Then, $E = \frac{1}{2} + \frac{1}{2} = 1$.

Case 2: $\omega$ is a primitive root, so $\omega^2 + \omega + 1 = 0$. Then, as shown earlier, $E = 1$.

Therefore, in all cases, $E = 1$. Therefore, the only possible value is 1.

Therefore, the answer is $\boxed{1}$.

**Final Answer**
\boxed{1}

To evaluate the expression
$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}
$$
where $\omega$ is a complex number such that $\omega^3 = 1$, we begin by identifying the possible values of $\omega$.

---

### Step 1: Identify the Roots of $\omega^3 = 1$

The equation $\omega^3 = 1$ has three roots, known as the **cube roots of unity**:

- $\omega = 1$
- $\omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$
- $\omega = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$

These roots satisfy the identity:
$$
1 + \omega + \omega^2 = 0
$$
This identity will be crucial in simplifying the expression.

---

### Step 2: Evaluate the Expression for Each Root

Let’s compute the expression for each of the three roots.

#### Case 1: $\omega = 1$

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = \frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1
$$

#### Case 2: $\omega$ is a non-real root (say, $\omega = e^{2\pi i / 3}$)

We use the identity $1 + \omega + \omega^2 = 0$, so $\omega^2 = -1 - \omega$. Then:

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = \frac{1}{1 + \omega} + \frac{1}{1 + (-1 - \omega)} = \frac{1}{1 + \omega} + \frac{1}{-\omega}
$$

Now simplify:

$$
\frac{1}{1 + \omega} - \frac{1}{\omega}
$$

Combine the terms:

$$
\frac{\omega - (1 + \omega)}{\omega(1 + \omega)} = \frac{-1}{\omega(1 + \omega)}
$$

Now compute the denominator:

$$
\omega(1 + \omega) = \omega + \omega^2 = -1
$$

So the expression becomes:

$$
\frac{-1}{-1} = 1
$$

#### Case 3: $\omega = \omega^2$ (the other non-real root)

This is symmetric to Case 2, and the same steps apply. The result is again:

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = 1
$$

---

### Step 3: Conclusion

In all three cases, the value of the expression is **1**. Therefore, regardless of which cube root of unity $\omega$ is, the expression always evaluates to the same value.

---

### Final Answer

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to find all possible values of the expression $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$ where $\omega$ is a complex number such that $\omega^3 = 1$. Hmm, let me start by recalling what the roots of the equation $\omega^3 = 1$ are.

First, I know that the solutions to $\omega^3 = 1$ are the cube roots of unity. There's one real root and two complex roots. The real root is obviously 1 because $1^3 = 1$. The other two roots are complex numbers, often denoted as $\omega$ and $\omega^2$, which are complex conjugates of each other.

So, the three cube roots of unity are 1, $\omega$, and $\omega^2$, where $\omega = e^{2\pi i /3}$ and $\omega^2 = e^{4\pi i /3}$. These satisfy the equation $1 + \omega + \omega^2 = 0$. That's a key identity I remember. Maybe that will come in handy later.

Now, the problem is asking for the value of $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me note that $\omega$ can be either of the complex roots or the real root 1. Wait, but if $\omega$ is 1, then the expression becomes $\frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1$. But the problem says "all possible values," so maybe there are different values depending on which root $\omega$ is? Wait, but the problem says "Let $\omega$ be a complex number such that $\omega^3 = 1$." So $\omega$ can be any of the three roots. However, if $\omega$ is 1, then the expression is 1. If $\omega$ is a complex root, then maybe the expression evaluates to something else?

Wait, but maybe the problem is considering $\omega$ as a non-real root? Because if $\omega$ is 1, then the expression is 1, but maybe the problem wants the values when $\omega$ is not 1? Wait, the problem says "Let $\omega$ be a complex number such that $\omega^3 = 1$", so $\omega$ can be any of the three roots. Therefore, we need to check all three possibilities. However, maybe when $\omega$ is 1, the expression is 1, and when $\omega$ is the other roots, maybe the expression is the same? Or different?

Wait, let me check. Let me first consider $\omega$ as the primitive root, so $\omega \neq 1$. Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Maybe I can combine these fractions. Let me try that.

First, find a common denominator. The denominators are $1 + \omega$ and $1 + \omega^2$. The common denominator would be $(1 + \omega)(1 + \omega^2)$. Let me compute the numerator:

$1 \cdot (1 + \omega^2) + 1 \cdot (1 + \omega) = (1 + \omega^2 + 1 + \omega) = 2 + \omega + \omega^2$.

So the combined fraction is $\frac{2 + \omega + \omega^2}{(1 + \omega)(1 + \omega^2)}$.

But from the identity $1 + \omega + \omega^2 = 0$, we know that $\omega + \omega^2 = -1$. Therefore, substituting that into the numerator:

$2 + (-1) = 1$. So the numerator simplifies to 1.

Now, let me compute the denominator: $(1 + \omega)(1 + \omega^2)$. Let me expand this:

$1 \cdot 1 + 1 \cdot \omega^2 + \omega \cdot 1 + \omega \cdot \omega^2 = 1 + \omega^2 + \omega + \omega^3$.

But $\omega^3 = 1$, so substituting that in:

$1 + \omega^2 + \omega + 1 = 2 + \omega + \omega^2$. Again, using the identity $\omega + \omega^2 = -1$, this becomes $2 + (-1) = 1$.

Therefore, the denominator is 1. Therefore, the entire expression simplifies to $\frac{1}{1} = 1$. Wait, so regardless of which root $\omega$ is (as long as it's not 1?), the expression is 1? But when $\omega = 1$, we saw that the expression is also 1. So does that mean that regardless of which cube root of unity we take, the value of the expression is always 1? Therefore, the only possible value is 1?

But that seems strange. Let me check with specific values. Let me take $\omega = e^{2\pi i /3}$. Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$.

First, compute $1 + \omega$. Since $\omega = e^{2\pi i /3} = -1/2 + i\sqrt{3}/2$, so $1 + \omega = 1 - 1/2 + i\sqrt{3}/2 = 1/2 + i\sqrt{3}/2$. Similarly, $\omega^2 = e^{4\pi i /3} = -1/2 - i\sqrt{3}/2$, so $1 + \omega^2 = 1 - 1/2 - i\sqrt{3}/2 = 1/2 - i\sqrt{3}/2$.

Therefore, the expression becomes $\frac{1}{1/2 + i\sqrt{3}/2} + \frac{1}{1/2 - i\sqrt{3}/2}$. Let me compute each term.

First term: $\frac{1}{1/2 + i\sqrt{3}/2}$. Multiply numerator and denominator by the conjugate of the denominator, which is $1/2 - i\sqrt{3}/2$:

$$
\frac{1 \cdot (1/2 - i\sqrt{3}/2)}{(1/2 + i\sqrt{3}/2)(1/2 - i\sqrt{3}/2)} = \frac{1/2 - i\sqrt{3}/2}{(1/2)^2 + (\sqrt{3}/2)^2} = \frac{1/2 - i\sqrt{3}/2}{1/4 + 3/4} = \frac{1/2 - i\sqrt{3}/2}{1} = 1/2 - i\sqrt{3}/2
$$

Similarly, the second term is $\frac{1}{1/2 - i\sqrt{3}/2}$. Multiply numerator and denominator by the conjugate $1/2 + i\sqrt{3}/2$:

$$
\frac{1 \cdot (1/2 + i\sqrt{3}/2)}{(1/2 - i\sqrt{3}/2)(1/2 + i\sqrt{3}/2)} = \frac{1/2 + i\sqrt{3}/2}{1} = 1/2 + i\sqrt{3}/2
$$

Adding the two terms together: $(1/2 - i\sqrt{3}/2) + (1/2 + i\sqrt{3}/2) = 1/2 + 1/2 + (-i\sqrt{3}/2 + i\sqrt{3}/2) = 1 + 0 = 1$.

So that checks out. So even when $\omega$ is a complex root, the expression is 1. And when $\omega = 1$, as we saw earlier, the expression is also 1. Therefore, regardless of which cube root of unity we take, the value of the expression is always 1. Therefore, the only possible value is 1.

Wait, but the problem says "Find all possible values," so maybe there's only one value? But maybe I made a mistake here? Let me check again.

Alternatively, maybe there's a different approach. Let me think. Since $\omega^3 = 1$, and $\omega \neq 1$, then $\omega$ satisfies $\omega^2 + \omega + 1 = 0$. So maybe I can use that to simplify the expression.

Let me try another approach. Let me denote the expression as $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me compute $E$:

First, note that $\omega^2 = \omega^{-1}$ because $\omega^3 = 1$, so $\omega^2 = \omega^{-1}$. Similarly, $1 + \omega^2 = 1 + \omega^{-1}$.

Alternatively, maybe I can write $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me combine them:

As before, common denominator is $(1 + \omega)(1 + \omega^2)$. The numerator is $(1 + \omega^2) + (1 + \omega) = 2 + \omega + \omega^2 = 2 - 1 = 1$, since $\omega + \omega^2 = -1$. The denominator is $(1 + \omega)(1 + \omega^2)$. Let me compute that again:

Expanding $(1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega^3 = 1 + (\omega + \omega^2) + \omega^3 = 1 + (-1) + 1 = 1$. Therefore, the denominator is 1, so $E = 1/1 = 1$.

Therefore, regardless of which root $\omega$ is (even if $\omega = 1$), the expression is 1. Wait, but if $\omega = 1$, then the denominator in the original expression would be $1 + 1 = 2$ for both terms, so $1/2 + 1/2 = 1$. So even in that case, it's 1. Therefore, regardless of which cube root of unity we take, the value is always 1. Therefore, the only possible value is 1.

But the problem says "Find all possible values," so maybe there's only one value? But maybe I need to check if there are different values? Wait, but according to all calculations, it's always 1.

Alternatively, maybe there's a mistake in assuming that $\omega$ can be 1? Let me check the problem statement again. It says "Let $\omega$ be a complex number such that $\omega^3 = 1$." So $\omega$ can be 1, or the other two roots. But in all cases, the expression evaluates to 1. Therefore, the only possible value is 1.

But maybe the problem is expecting different values? Wait, let me think again. Suppose $\omega$ is a primitive cube root of unity, so $\omega \neq 1$. Then, the expression is 1. If $\omega = 1$, then the expression is also 1. Therefore, regardless of which root we take, the value is 1. Therefore, the only possible value is 1.

Alternatively, maybe there's a different interpretation? For example, maybe the problem is considering $\omega$ as a non-real root? But even then, the expression still evaluates to 1.

Alternatively, maybe I made a mistake in the algebra? Let me check again.

Let me take $\omega = 1$. Then, $\frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1$.

If $\omega$ is a non-real root, then as shown earlier, the expression is 1. Therefore, regardless of which root we take, the value is 1. Therefore, the only possible value is 1.

But maybe the problem is expecting multiple values? Wait, but according to the math, it's always 1. Let me check with another approach. Let me consider that $\omega^3 = 1$, so $\omega^3 - 1 = 0$, which factors as $(\omega - 1)(\omega^2 + \omega + 1) = 0$. Therefore, either $\omega = 1$ or $\omega^2 + \omega + 1 = 0$.

So, if $\omega \neq 1$, then $\omega^2 + \omega + 1 = 0$. Therefore, $\omega^2 = -1 - \omega$. Let me substitute this into the expression.

Let me compute $\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Since $\omega^2 = -1 - \omega$, then $1 + \omega^2 = 1 + (-1 - \omega) = -\omega$. Therefore, the second term becomes $\frac{1}{1 + \omega^2} = \frac{1}{-\omega} = -\frac{1}{\omega}$.

Therefore, the entire expression becomes $\frac{1}{1 + \omega} - \frac{1}{\omega}$. Let me compute this.

First, let me write $\frac{1}{1 + \omega} - \frac{1}{\omega}$. Let me combine these terms:

$$
\frac{\omega - (1 + \omega)}{\omega(1 + \omega)} = \frac{\omega - 1 - \omega}{\omega(1 + \omega)} = \frac{-1}{\omega(1 + \omega)}
$$

Hmm, so that's $\frac{-1}{\omega(1 + \omega)}$. Let me compute the denominator: $\omega(1 + \omega) = \omega + \omega^2$. But since $\omega^2 = -1 - \omega$, then $\omega + \omega^2 = \omega + (-1 - \omega) = -1$. Therefore, the denominator is $-1$, so the entire expression becomes $\frac{-1}{-1} = 1$.

So again, we get 1. Therefore, regardless of whether $\omega$ is 1 or a non-real root, the expression is 1. Therefore, the only possible value is 1.

Therefore, the answer is just 1.

But the problem says "Find all possible values," so maybe there's only one value? Maybe the problem is designed to show that even though there are three roots, the expression evaluates to the same value.

Alternatively, maybe I need to check if there's a case where the expression is undefined? For example, if $1 + \omega = 0$ or $1 + \omega^2 = 0$. But if $1 + \omega = 0$, then $\omega = -1$. But $\omega^3 = (-1)^3 = -1 \neq 1$, so $\omega = -1$ is not a root of $\omega^3 = 1$. Therefore, the denominators are never zero. Therefore, the expression is defined for all cube roots of unity.

Therefore, the conclusion is that the expression always equals 1. Therefore, the only possible value is 1.

But just to be thorough, let me check with another method. Let me consider that $\omega$ is a root of $x^3 - 1 = 0$, so $x^3 = 1$. Let me consider the expression $E = \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}$. Let me compute $E$ for each root:

Case 1: $\omega = 1$. Then, $E = \frac{1}{2} + \frac{1}{2} = 1$.

Case 2: $\omega$ is a primitive root, so $\omega^2 + \omega + 1 = 0$. Then, as shown earlier, $E = 1$.

Therefore, in all cases, $E = 1$. Therefore, the only possible value is 1.

Therefore, the answer is $\boxed{1}$.

**Final Answer**
\boxed{1}

To evaluate the expression
$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}
$$
where $\omega$ is a complex number such that $\omega^3 = 1$, we begin by identifying the possible values of $\omega$.

---

### Step 1: Identify the Roots of $\omega^3 = 1$

The equation $\omega^3 = 1$ has three roots, known as the **cube roots of unity**:

- $\omega = 1$
- $\omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$
- $\omega = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$

These roots satisfy the identity:
$$
1 + \omega + \omega^2 = 0
$$
This identity will be crucial in simplifying the expression.

---

### Step 2: Evaluate the Expression for Each Root

Let’s compute the expression for each of the three roots.

#### Case 1: $\omega = 1$

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = \frac{1}{1 + 1} + \frac{1}{1 + 1} = \frac{1}{2} + \frac{1}{2} = 1
$$

#### Case 2: $\omega$ is a non-real root (say, $\omega = e^{2\pi i / 3}$)

We use the identity $1 + \omega + \omega^2 = 0$, so $\omega^2 = -1 - \omega$. Then:

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = \frac{1}{1 + \omega} + \frac{1}{1 + (-1 - \omega)} = \frac{1}{1 + \omega} + \frac{1}{-\omega}
$$

Now simplify:

$$
\frac{1}{1 + \omega} - \frac{1}{\omega}
$$

Combine the terms:

$$
\frac{\omega - (1 + \omega)}{\omega(1 + \omega)} = \frac{-1}{\omega(1 + \omega)}
$$

Now compute the denominator:

$$
\omega(1 + \omega) = \omega + \omega^2 = -1
$$

So the expression becomes:

$$
\frac{-1}{-1} = 1
$$

#### Case 3: $\omega = \omega^2$ (the other non-real root)

This is symmetric to Case 2, and the same steps apply. The result is again:

$$
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} = 1
$$

---

### Step 3: Conclusion

In all three cases, the value of the expression is **1**. Therefore, regardless of which cube root of unity $\omega$ is, the expression always evaluates to the same value.

---

### Final Answer

$$
\boxed{1}
$$

Solution:
We can write
\begin{align*}
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} &= \frac{1 + \omega^2 + 1 + \omega}{(1 + \omega)(1 + \omega^2)} \\
&= \frac{2 + \omega + \omega^2}{1 + \omega + \omega^2 + \omega^3} \\
&= \frac{2 + \omega + \omega^2}{2 + \omega + \omega^2} \\
&= \boxed{1}.
\end{align*}