Question

sum of cubes:
$(a + b)(a^2 - ab + b^2) = a^3 + b^3$
difference of cubes:
$(a - b)(a^2 + ab + b^2) = a^3 - b^3$
which products result in a sum or difference of cubes?
check all that apply.
$square$ $(x - 4)(x^2 + 4x - 16)$
$square$ $(x - 1)(x^2 - x + 1)$
$square$ $(x - 1)(x^2 + x + 1)$
$square$ $(x + 1)(x^2 + x - 1)$
$square$ $(x + 4)(x^2 - 4x + 16)$
$square$ $(x + 4)(x^2 + 4x + 16)$

Explanation:

Step1: Match to difference of cubes

Compare each option to $(a-b)(a^2+ab+b^2)=a^3-b^3$.
For $(x-1)(x^2+x+1)$: $a=x$, $b=1$. Substitute:
$(x-1)(x^2+(x)(1)+1^2)=x^3-1^3=x^3-1$, which fits the difference of cubes formula.

Step2: Match to sum of cubes

Compare each option to $(a+b)(a^2-ab+b^2)=a^3+b^3$.
For $(x+4)(x^2-4x+16)$: $a=x$, $b=4$. Substitute:
$(x+4)(x^2-(x)(4)+4^2)=x^3+4^3=x^3+64$, which fits the sum of cubes formula.

Step3: Verify other options

• $(x-4)(x^2+4x-16)$: Last term is $-16

eq4^2$, does not fit.

• $(x-1)(x^2-x+1)$: Middle term is $-x

eq+(x)(1)$, does not fit.

• $(x+1)(x^2+x-1)$: Last term is $-1

eq1^2$, does not fit.

• $(x+4)(x^2+4x+16)$: Middle term is $+4x

eq-(x)(4)$, does not fit.

Answer:

• $(x-1)(x^2+x+1)$
• $(x+4)(x^2-4x+16)$