Question

a student factors $a^6 - 64$ to $(a^2 - 4)(a^4 + 4a^2 + 16)$.
which statement about $(a^2 - 4)(a^4 + 4a^2 + 16)$ is correct?
○ the expression is equivalent and is completely factored.
○ the expression is equivalent, but the $(a^2 - 4)$ term is not completely factored.
○ the expression is equivalent, but the $(a^4 + 4a^2 + 16)$ term is not completely factored.
○ the expression is not equivalent.
done

Explanation:

Step1: Verify equivalence via formula

Recall difference of cubes: $x^3 - y^3=(x-y)(x^2+xy+y^2)$. Let $x=a^2$, $y=4$.
$$a^6 - 64=(a^2)^3 - 4^3=(a^2-4)(a^4+4a^2+16)$$

Step2: Check full factorization

Factor $a^2-4$ (difference of squares):
$$a^2-4=(a-2)(a+2)$$
$a^4+4a^2+16$ cannot be factored over integers.

Answer:

The expression is equivalent, but the $(a^{2}-4)$ term is not completely factored.