Question

we the distributive property to verify which expression is equivalent to (x-5) (x-3).
A. 3x²-4x²-3x-15
B. 2x-2
C. X²-8x+15
D. X²-2x+15

Answer 1

To verify which expression is equivalent to \((x-5)(x-3)\), we can use the distributive property (also known as the FOIL method for binomials):

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

Calculating each part step-by-step:
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 these parts:

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

So, \((x-5)(x-3)\) simplifies to \(x^2 - 8x + 15\).

Now we can compare this with the provided options:

A. \(3x^2 - 4x^2 - 3x - 15\)

Simplifying A:
\[
3x^2 - 4x^2 - 3x - 15 = -x^2 - 3x - 15 \quad \text{(not equivalent)}
\]

B. \(2x - 2\) (not a polynomial of the same degree; not equivalent)

C. \(x^2 - 8x + 15\) (this matches what we calculated; **this is equivalent**)

D. \(x^2 - 2x + 15\) (not equivalent)

The expression that is equivalent to \((x-5)(x-3)\) is:

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