To prove that the expressions \((x-5)(x+3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent using the distributive property, we will expand expression 1 and simplify expression 2.
Expression 1: \((x-5)(x+3)\)
Using the distributive property (also known as the FOIL method for binomials):
\[ (x-5)(x+3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]
Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)
Now simplifying expression 2 step-by-step:
\[ 2x^2 - x^2 = x^2 \] So we can rewrite it as: \[ x^2 - 2x - 20 + 5 \] Next, simplify the constants: \[ -20 + 5 = -15 \] So, we have: \[ x^2 - 2x - 15 \]
Now we have both expressions simplified:
- Expression 1: \(x^2 - 2x - 15\)
- Expression 2: \(x^2 - 2x - 15\)
Since both expressions simplify to the same polynomial \(x^2 - 2x - 15\), we conclude that they are equivalent.
Final Answer: Both expressions represent the same polynomial, confirming their equivalence through the distributive property applied to simplify and confirm both yield \(x^2 - 2x - 15\).
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