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: 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.