Select the correct answer.

What is the completely factored form of this polynomial?

[tex]\[ 2x^5 + 12x^3 - 54x \][/tex]

A. [tex]\[ 2x\left(x^2+3\right)(x+9)(x-9) \][/tex]

B. [tex]\[ 2x(x-3)(x+9) \][/tex]

C. [tex]\[ 2x\left(x^2+3\right)(x+3)(x-3) \][/tex]

D. [tex]\[ 2x\left(x^2-3\right)\left(x^2+9\right) \][/tex]



Answer :

To factor the polynomial [tex]\(2x^5 + 12x^3 - 54x\)[/tex] completely, let us follow these steps:

1. Factor out the greatest common factor (GCF):
The GCF of the terms [tex]\(2x^5\)[/tex], [tex]\(12x^3\)[/tex], and [tex]\(-54x\)[/tex] is [tex]\(2x\)[/tex]. So, we can factor out [tex]\(2x\)[/tex] from the polynomial.

[tex]\[ 2x^5 + 12x^3 - 54x = 2x (x^4 + 6x^2 - 27) \][/tex]

2. Factor the remaining polynomial [tex]\(x^4 + 6x^2 - 27\)[/tex]:

By setting [tex]\(u = x^2\)[/tex], we can rewrite [tex]\(x^4 + 6x^2 - 27\)[/tex] as a quadratic in terms of [tex]\(u\)[/tex]:
[tex]\[ u^2 + 6u - 27 \][/tex]

We now need to factor [tex]\(u^2 + 6u - 27\)[/tex]. This can be done by finding two numbers that multiply to [tex]\(-27\)[/tex] and add to [tex]\(6\)[/tex]. Those numbers are [tex]\(9\)[/tex] and [tex]\(-3\)[/tex].

Thus, we can write:
[tex]\[ u^2 + 6u -27 = (u + 9)(u - 3) \][/tex]

Since [tex]\(u = x^2\)[/tex], we substitute back to get:
[tex]\[ (x^2 + 9)(x^2 - 3) \][/tex]

3. Combine everything back together:
Substituting these factors back into our factored expression and including the GCF we factored out earlier:
[tex]\[ 2x (x^2 + 9)(x^2 - 3) \][/tex]

Thus, the completely factored form is:
[tex]\[ 2x (x^2 - 3)(x^2 + 9) \][/tex]

Now we compare the result with the given options:

A. [tex]\(2x (x^2 + 3)(x + 9)(x - 9)\)[/tex]

B. [tex]\(2x (x - 3)(x + 9)\)[/tex]

C. [tex]\(2x (x^2 + 3)(x + 3)(x - 3)\)[/tex]

D. [tex]\(2x (x^2 - 3)(x^2 + 9)\)[/tex]

The correct answer is:
[tex]\[ \boxed{D} \][/tex]