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: 2x2−x2−2x−20+5 (4 points)

Answer 1

To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we'll simplify both expressions and show that they yield the same result.

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

Using the Distributive Property (also known as the FOIL method for binomials):
\[
(x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3
\]
Calculating each term:
\[
= x^2 + 3x - 5x - 15
\]
Combining like terms:
\[
= x^2 - 2x - 15
\]

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

Combine like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
Calculating the first part:
\[
= x^2 - 2x - 20 + 5
\]
Then combining the constants:
\[
= x^2 - 2x - 15
\]

**Conclusion**:

Both expressions simplify to \(x^2 - 2x - 15\). Therefore, we conclude that:
- **Expression 1, \((x - 5)(x + 3)\), simplifies to \(x^2 - 2x - 15\) using the Distributive Property.**
- **Expression 2, \(2x^2 - x^2 - 2x - 20 + 5\), also simplifies to \(x^2 - 2x - 15\) upon combining like terms.**

Since both expressions are equal, we have proved that they are equivalent.