Use the Distributive Property to show that these algebraic expressions are equivalent.

Question

Expression 1: (x−5)(x+3)


Expression 2: 2x2−x2−2x−20+5


Explain your answer in one sentence for each expression (2 pts)
Show the steps of your work for each expression (2 pts)


(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 on Expression 1 and simplify Expression 2.

Expression 1: \((x−5)(x+3)\)

Explanation: The expression is equivalent to the sum of the products of each term in the first binomial with each term in the second binomial.

Steps:

1. Distribute \(x\) to both terms in \((x + 3)\): \[ x \cdot (x + 3) = x^2 + 3x \]
2. Distribute \(-5\) to both terms in \((x + 3)\): \[ -5 \cdot (x + 3) = -5x - 15 \]
3. Combine the results: \[ x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \]

Expression 2: \(2x^2−x^2−2x−20+5\)

Explanation: This expression is equivalent to performing standard operations, combining like terms to simplify.

Steps:

1. Combine \(2x^2\) and \(-x^2\): \[ 2x^2 - x^2 = x^2 \]
2. Combine the constant terms: \[ -20 + 5 = -15 \]
3. Combine all terms: \[ x^2 - 2x - 15 \]

Conclusion

Both expressions simplify to \(x^2 - 2x - 15\), thus proving that \((x−5)(x+3)\) is equivalent to \(2x^2−x^2−2x−20+5\).