Question

use the distributive property to prove that these algebraic expressions are equivalent explain your answer in one sentence for each expression and show your work

expression 1: (x-5)(x+3)

expression 2: 2x^2 -x^2 - 2x - 20 + 5

Answer 1

To prove that the expressions \((x-5)(x+3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent using the distributive property, we will expand expression 1 and simplify expression 2.

**Expression 1: \((x-5)(x+3)\)**

Using the distributive property (also known as the FOIL method for binomials):

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

**Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)**

Now simplifying expression 2 step-by-step:

\[
2x^2 - x^2 = x^2
\]
So we can rewrite it as:
\[
x^2 - 2x - 20 + 5
\]
Next, simplify the constants:
\[
-20 + 5 = -15
\]
So, we have:
\[
x^2 - 2x - 15
\]

Now we have both expressions simplified:
1. Expression 1: \(x^2 - 2x - 15\)
2. Expression 2: \(x^2 - 2x - 15\)

Since both expressions simplify to the same polynomial \(x^2 - 2x - 15\), we conclude that they are equivalent.

**Final Answer:**
Both expressions represent the same polynomial, confirming their equivalence through the distributive property applied to simplify and confirm both yield \(x^2 - 2x - 15\).