Question

1. identify all correct factorizations for $512x^{3}+0.343$:

a. $(8x - 0.7)(64x^{2}-5.6x - 0.49)$
b. $(8x + 0.7)(64x^{2}-5.6x - 0.49)$
c. $(8x - 0.7)(64x^{2}+5.6x - 0.49)$
d. $(8x + 0.7)(64x^{2}-5.6x + 0.49)$

Explanation:

Step1: Recognize sum of cubes

Recall $a^3+b^3=(a+b)(a^2-ab+b^2)$

Step2: Match terms to formula

$512x^3=(8x)^3$, $0.343=(0.7)^3$, so $a=8x$, $b=0.7$

Step3: Compute the quadratic factor

$a^2-ab+b^2=(8x)^2-(8x)(0.7)+(0.7)^2=64x^2-5.6x+0.49$

Step4: Combine factors

$(8x+0.7)(64x^2-5.6x+0.49)$

Answer:

d. $(8x + 0.7)(64x^2 - 5.6x + 0.49)$