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 two expressions are equivalent using the Distributive Property, we will first simplify each expression step by step.

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

We will apply the Distributive Property (FOIL method) to multiply the two binomials:

1. First: \(x \cdot x = x^2\)
2. Outer: \(x \cdot 3 = 3x\)
3. Inner: \(-5 \cdot x = -5x\)
4. Last: \(-5 \cdot 3 = -15\)

Now, combine the like terms:

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

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

Now, simplify this expression by combining like terms:

1. Combine \(2x^2 - x^2\):
\[
2x^2 - x^2 = x^2
\]
2. Next, combine the constants \(-20 + 5\):
\[
-20 + 5 = -15
\]

Putting it all together, we have:

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

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**Conclusion:**

After simplifying both expressions, we find that:

- Expression 1 simplifies to \(x^2 - 2x - 15\).
- Expression 2 simplifies to \(x^2 - 2x - 15\).

Since both expressions result in the same polynomial, we conclude that the expressions are equivalent.