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 first expand Expression 1 and then simplify Expression 2.
Step 1: Expand Expression 1
Using the Distributive Property (also known as the FOIL method for binomials), we have:
\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]
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:
Now we see that both expressions simplify to the same form:
Expression 1: \(x^2 - 2x - 15\)
Expression 2: \(x^2 - 2x - 15\)
Thus, \((x−5)(x+3) = 2x^2−x^2−2x−20+5\) are equivalent since they yield the same polynomial expression when simplified.
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