Question

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 first expand Expression 1 and then simplify Expression 2.

**Step 1: Expand Expression 1**
Using the Distributive Property (also known as the FOIL method for binomials), we have:

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

**Step 2: Simplify Expression 2**
Now let's simplify Expression 2:

\[
2x^2 - x^2 - 2x - 20 + 5
\]
Combine like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
\[
= x^2 - 2x - 15
\]

**Conclusion:**
Now we see that both expressions simplify to the same form:
Expression 1: \(x^2 - 2x - 15\)
Expression 2: \(x^2 - 2x - 15\)

Thus, \((x−5)(x+3) = 2x^2−x^2−2x−20+5\) are equivalent since they yield the same polynomial expression when simplified.