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. expression1 (x-5)(x+3) expression2 2x^2-x^2-2x-20+5

Answer 1

To prove that the expressions (x - 5)(x + 3) and 2x² - x² - 2x - 20 + 5 are equivalent using the Distributive Property, we first need to simplify both sides.

Step 1: Expand Expression 1 \[ (x - 5)(x + 3) = x(x + 3) - 5(x + 3) \quad \text{(using the Distributive Property)} \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]

Step 2: Simplify Expression 2 Next, we simplify expression 2: \[ 2x^2 - x^2 - 2x - 20 + 5 \] Combining like terms: \[ = (2x^2 - x^2) - 2x - 20 + 5 \] \[ = x^2 - 2x - 15 \]

Conclusion: Now we see that both simplified expressions are equal: \[ x^2 - 2x - 15 = x^2 - 2x - 15 \] Thus, the expressions (x - 5)(x + 3) and 2x² - x² - 2x - 20 + 5 are equivalent because they simplify to the same polynomial expression.

Answer 2

can you put this in two sentences were one sentec is the first expression and the other sentences is the second expression

Answer 3

The expression \((x - 5)(x + 3)\) simplifies to \(x^2 - 2x - 15\) using the Distributive Property. The expression \(2x^2 - x^2 - 2x - 20 + 5\) also simplifies to \(x^2 - 2x - 15\), demonstrating that both expressions are equivalent.