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 squared minus x squared minus 2X - 20 + 5

Answer 1

To prove that the two algebraic expressions are equivalent using the distributive property, we'll simplify both expressions step by step.

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

1. Distributing \(-5\) in the expression: \[ x - 5(x + 3) = x - 5x - 15 \]
2. Combine like terms: \[ x - 5x - 15 = -4x - 15 \]

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

1. Combine \(2x^2\) and \(-x^2\): \[ 2x^2 - x^2 = x^2 \]
2. Now rewrite the expression: \[ x^2 - 2x - 20 + 5 \]
3. Combine the constants \(-20\) and \(5\): \[ -20 + 5 = -15 \]
4. Final expression: \[ x^2 - 2x - 15 \]

Now, let's rewrite both simplified expressions:

• Expression 1 simplified is \(-4x - 15\).
• Expression 2 simplified is \(x^2 - 2x - 15\).

Conclusion:

The expressions \(x - 5(x + 3)\) simplifies to \(-4x - 15\), and \(2x^2 - x^2 - 2x - 20 + 5\) simplifies to \(x^2 - 2x - 15\). Since there are different degrees (\(-4x - 15\) is linear while \(x^2 - 2x - 15\) is quadratic), these expressions are not equivalent. Thus, they represent distinct expressions in algebra.