Question

use the distributive property to verify expression is equivalent to (x-5)(x-3).
x^2-2x+15
3x^2-4x^2-3x-15
2x-2
x^2-8x+15

Answer 1

Let's use the distributive property to expand the expression \((x-5)(x-3)\) and then check which of the given expressions is equivalent to it.

First, let's expand \((x-5)(x-3)\) using the distributive property (also known as FOIL for binomials):

\[
(x-5)(x-3) = x \cdot x + x \cdot (-3) + (-5) \cdot x + (-5) \cdot (-3)
\]

Calculating each term:

- \(x \cdot x = x^2\)
- \(x \cdot (-3) = -3x\)
- \((-5) \cdot x = -5x\)
- \((-5) \cdot (-3) = +15\)

Now, add them together:

\[
x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]

Now, we've determined that:

\[
(x-5)(x-3) = x^2 - 8x + 15
\]

Next, let's check which of the given expressions are equivalent to \(x^2 - 8x + 15\):

1. **For \(x^2 - 2x + 15\):**
- This is not equivalent because the middle term \(-2x\) is different.

2. **For \(3x^2 - 4x^2 - 3x - 15\):**
- Simplifying \(3x^2 - 4x^2 - 3x - 15\) gives \(-x^2 - 3x - 15\).
- This is also not equivalent to \(x^2 - 8x + 15\).

3. **For \(2x - 2\):**
- This is not equivalent either, as it is simply a linear expression, while we have a quadratic with mixed terms.

4. **For \(x^2 - 8x + 15\):**
- This matches exactly with our result from expanding \((x-5)(x-3)\).

Thus, from the given options, the equivalent expression to \((x-5)(x-3)\) is:

\[
x^2 - 8x + 15
\]