Question

Structures of Expressions Unit Test:
Use the Distributive Property to prove that these algebraic expression are equivalents. Explain your answer in one sentence for each expression and show your work.
Expression 1: (x - 5) (x + 3)
Expression 2: 2 x2 - x2 - 2x- 20 + 5

Answer 1

To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we can simplify both expressions and show that they are equal.

### Work for Expression 1:
1. **Apply the Distributive Property (FOIL method)**:
\[
(x - 5)(x + 3) = x(x) + x(3) - 5(x) - 5(3)
\]
\[
= x^2 + 3x - 5x - 15
\]
\[
= x^2 - 2x - 15
\]

### Work for Expression 2:
1. **Simplify the expression**:
Combine like terms in \(2x^2 - x^2 - 2x - 20 + 5\):
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
\[
= x^2 - 2x - 15
\]

### Conclusion:
Both expressions simplify to \(x^2 - 2x - 15\), thus proving that \((x - 5)(x + 3)\) is equivalent to \(2x^2 - x^2 - 2x - 20 + 5\) because they yield the same polynomial when fully simplified.