Question

Use the Distributive Property to verify which expression is equivalent to (x–5)(x–3).

A. 3x^2–4x^2–3x–15
B. 2x–2
C. X^2–2x+15
D. X^2–8x+15

Answer 1

To verify which expression is equivalent to \((x - 5)(x - 3)\) using the Distributive Property, we will expand the expression:

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

Now let's perform the multiplication:

1. \(x \cdot x = x^2\)
2. \(x \cdot (-3) = -3x\)
3. \((-5) \cdot x = -5x\)
4. \((-5) \cdot (-3) = 15\)

Now, we can combine all the terms:

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

So, \((x - 5)(x - 3)\) simplifies to:

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

Now, we can compare this result to the given options:

A. \(3x^2 - 4x^2 - 3x - 15\) = \( -x^2 - 3x - 15\) (not equivalent)
B. \(2x - 2\) (not equivalent)
C. \(x^2 - 2x + 15\) (not equivalent)
D. \(x^2 - 8x + 15\) (equivalent)

Thus, the correct answer is:

**D. \(x^2 - 8x + 15\)**