Question

which expression is equivalent to factoring the difference of cubes expression ( x^3 - 8 )?

a. ( (x - 2)(x^2 - 2x - 4) )

b. ( (x + 2)(x^2 + 2x - 4) )

c. ( (x + 2)(x^2 - 2x + 4) )

d. ( (x - 2)(x^2 + 2x + 4) )

what is the factored form of ( 512x^3 - 343 )?

a. ( (8x - 7)(64x^2 + 56x + 49) )

b. ( (8x + 7)(64x^2 - 56x + 49) )

c. ( (8x - 7)(64x^2 - 56x - 49) )

d. ( (8x + 7)(64x^2 + 56x + 49) )

Explanation:

Step1: Recall difference of cubes formula

The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Step2: Solve first problem

For $x^3 - 8$, $a=x$, $b=2$ (since $2^3=8$).
Substitute into formula:

$$\begin{align*} x^3 - 8&=(x - 2)(x^2 + (x)(2) + 2^2)\\ &=(x - 2)(x^2 + 2x + 4) \end{align*}$$

Step3: Solve second problem

For $512x^3 - 343$, $a=8x$ (since $(8x)^3=512x^3$), $b=7$ (since $7^3=343$).
Substitute into formula:

$$\begin{align*} 512x^3 - 343&=(8x - 7)((8x)^2 + (8x)(7) + 7^2)\\ &=(8x - 7)(64x^2 + 56x + 49) \end{align*}$$

Answer:

1. d. $(x - 2)(x^2 + 2x + 4)$
2. a. $(8x - 7)(64x^2 + 56x + 49)$