Structures of Expressions Unit Test Part 2

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

Answer 1

To prove that the expressions \((x-5)(x+3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we can expand Expression 1 using the Distributive Property and simplify Expression 2.

Step 1: Expand Expression 1

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

Using the Distributive Property (also known as the FOIL method for binomials):

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

Calculating each term:

• \(x \cdot x = x^2\)
• \(x \cdot 3 = 3x\)
• \(-5 \cdot x = -5x\)
• \(-5 \cdot 3 = -15\)

Combining these terms:

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

Step 2: Simplify Expression 2

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

Combining like terms:

1. \(2x^2 - x^2 = x^2\)
2. The linear term is \(-2x\).
3. The constant terms are \(-20 + 5 = -15\).

So, we get:

\[ x^2 - 2x - 15 \]

Conclusion

Both expressions simplify to \(x^2 - 2x - 15\), therefore they are equivalent.

Answer Summary:

• Expression 1 expands to \(x^2 - 2x - 15\) using the Distributive Property.
• Expression 2 simplifies to \(x^2 - 2x - 15\) by combining like terms.
  Thus, both expressions are equivalent as they yield the same polynomial.