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]
