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.

Question

Expression 1: (x−5)(x+3)

Expression 2: 2x2−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 using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Step 1: Expanding Expression 1

We will use the Distributive Property (also known as the FOIL method for binomials) to expand \((x - 5)(x + 3)\):

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

Step 2: Simplifying Expression 2

Now let's simplify Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\):

\[ 2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x + (-20 + 5) \] \[ = x^2 - 2x - 15 \]

Conclusion

Both expressions simplify to \(x^2 - 2x - 15\), proving that \((x - 5)(x + 3) = 2x^2 - x^2 - 2x - 20 + 5\). Thus, we conclude that the two expressions are equivalent since both yield the same polynomial after simplification.