Question

the polynomial $x^3 + 8$ is equal to
$(x - 2)(x^2 + 2x + 8)$.
$(x + 2)(x^2 - 2x + 8)$.
$(x + 2)(x^2 - 2x + 4)$.
$(x - 2)(x^2 + 2x + 4)$

Explanation:

Step1: Identify sum of cubes

The polynomial $x^3 + 8$ is a sum of cubes, where $x^3 + 2^3$ (since $8=2^3$). The sum of cubes formula is $a^3 + b^3=(a+b)(a^2-ab+b^2)$.

Step2: Apply sum of cubes formula

Substitute $a=x$, $b=2$ into the formula:
$$(x+2)(x^2 - x\cdot2 + 2^2)$$
Simplify the second factor:
$$(x+2)(x^2 - 2x + 4)$$

Step3: Verify by expanding

Expand $(x+2)(x^2 - 2x + 4)$:
$$x(x^2 - 2x + 4) + 2(x^2 - 2x + 4)$$
$$=x^3 - 2x^2 + 4x + 2x^2 - 4x + 8$$
Combine like terms:
$$x^3 + ( -2x^2 + 2x^2) + (4x - 4x) + 8 = x^3 + 8$$

Answer:

$(x + 2)(x^2 - 2x + 4)$