Question

1. (factor) $-16x^5 - 250x^2$

$f(x) = 21x^4 - 84x^3 + 15x^2 - 60x$

Explanation:

Step1: Factor first expression's GCF

Identify greatest common factor (GCF) of $-16x^5$ and $-250x^2$, which is $-2x^2$.

$$\begin{align} -16x^5 -250x^2 &= -2x^2(8x^3 + 125) \end{align}$$

Step2: Factor sum of cubes

Recognize $8x^3+125$ as sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$ where $a=2x$, $b=5$.

$$\begin{align} 8x^3 + 125 &= (2x+5)((2x)^2-(2x)(5)+5^2) \\ &= (2x+5)(4x^2-10x+25) \end{align}$$

Step3: Combine first expression factors

$$\begin{align} -16x^5 -250x^2 &= -2x^2(2x+5)(4x^2-10x+25) \end{align}$$

Step4: Factor $f(x)$'s GCF

Identify GCF of $21x^4, -84x^3, 15x^2, -60x$, which is $3x$.

$$\begin{align} f(x) &= 3x(7x^3 -28x^2 +5x -20) \end{align}$$

Step5: Factor by grouping

Group terms in the cubic and factor each group.

$$\begin{align} 7x^3 -28x^2 +5x -20 &= 7x^2(x-4) +5(x-4) \\ &= (x-4)(7x^2+5) \end{align}$$

Step6: Combine $f(x)$ factors

$$\begin{align} f(x) &= 3x(x-4)(7x^2+5) \end{align}$$

Answer:

$-16x^5 -250x^2 = -2x^2(2x+5)(4x^2-10x+25)$
$f(x) = 3x(x-4)(7x^2+5)$