Question

select the correct answer.
what is the completely factored form of this polynomial?
$2x^5 + 12x^3 - 54x$
a. $2x(x^2 + 3)(x + 9)(x - 9)$
b. $2x(x - 3)(x + 9)$
c. $2x(x^2 + 3)(x + 3)(x - 3)$
d. $2x(x^2 - 3)(x^2 + 9)$

Explanation:

Step1: Factor out GCF

First, identify and factor out the greatest common factor (GCF) of all terms. The GCF of $2x^5$, $12x^3$, and $-54x$ is $2x$.
$2x^5 + 12x^3 - 54x = 2x(x^4 + 6x^2 - 27)$

Step2: Factor quadratic in $x^2$

Treat $y = x^2$ to rewrite the quartic as a quadratic: $y^2 + 6y - 27$. Factor this quadratic by finding two numbers that multiply to $-27$ and add to $6$ (these numbers are $9$ and $-3$).
$x^4 + 6x^2 - 27 = (x^2 + 9)(x^2 - 3)$

Step3: Factor difference of squares

Notice that $x^2 - 3$ is a difference of squares ($a^2 - b^2 = (a+b)(a-b)$ where $a=x$, $b=\sqrt{3}$), and $x^2 + 9$ cannot be factored over real numbers.
$x^2 - 3 = (x + 3)(x - 3)$

Step4: Combine all factors

Substitute back into the expression from Step1.
$2x(x^2 + 9)(x + 3)(x - 3) = 2x(x^2 + 3)(x + 3)(x - 3)$

Answer:

C. $2x(x^2 + 3)(x + 3)(x - 3)$