Structures of Expressions Unit Test

Question

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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: 2x2−x2−2x−20+5


(4 points)

Answer 1

To show that the expressions \((x-5)(x+3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we will use the Distributive Property to expand Expression 1 and simplify Expression 2.

Step 1: Expand Expression 1

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

Step 2: Simplify Expression 2

Now, let's simplify Expression 2: \[ 2x^2 - x^2 - 2x - 20 + 5 \] Combine like terms: \[ = (2x^2 - x^2) - 2x - 20 + 5 \] \[ = x^2 - 2x - 15 \]

Conclusion

After performing the calculations, we find that both expressions simplify to \(x^2 - 2x - 15\). Thus, we can conclude that the two expressions are equivalent.

Answer: Expression 1 simplifies to \(x^2 - 2x - 15\) and Expression 2 simplifies to \(x^2 - 2x - 15\), proving they are equivalent; both expressions ultimately equal \(x^2 - 2x - 15\).