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Find the binomial that completes the factorization. $\newline$ $q^3 + 64 = (\underline{\hspace{3em}})(q^2 - 4q + 16)$

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Factor sums and differences of cubes

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Q. Find the binomial that completes the factorization.

\newline

q^3 + 64 = (\underline{\hspace{3em}})(q^2 - 4q + 16)

Recognize Perfect Cube: Recognize that $64$ is a perfect cube, $64 = 4^3$ .
Use Sum of Cubes Formula: Use the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
Identify $a$ and $b$ : Identify $a = q$ and $b = 4$ , since $q^3 + 64$ can be written as $q^3 + 4^3$ .
Plug into Formula: Plug $a$ and $b$ into the sum of cubes formula: $q^3 + 4^3 = (q + 4)(q^2 - q\cdot 4 + 4^2)$ .
Simplify Binomial and Trinomial: Simplify the binomial and trinomial: $(q + 4)(q^2 - 4q + 16)$ .

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