To determine the completely factored form of the polynomial [tex]\(2x^5 + 12x^3 - 54x\)[/tex], let's go through the steps systematically:
1. Extract the Greatest Common Factor (GCF):
The given polynomial is [tex]\(2x^5 + 12x^3 - 54x\)[/tex].
The GCF of the coefficients [tex]\(2, 12,\)[/tex] and [tex]\(-54\)[/tex] is [tex]\(2\)[/tex]. For the variable part, the GCF is [tex]\(x\)[/tex]. Hence, we factor out [tex]\(2x\)[/tex] from the polynomial:
[tex]\[
2x(x^4 + 6x^2 - 27)
\][/tex]
2. Factor the Quartic Polynomial [tex]\(x^4 + 6x^2 - 27\)[/tex]:
We can think of [tex]\(x^4 + 6x^2 - 27\)[/tex] as a quadratic polynomial in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex], then the polynomial becomes:
[tex]\[
y^2 + 6y - 27
\][/tex]
3. Factor the Quadratic Polynomial:
We need to find two numbers that multiply to [tex]\(-27\)[/tex] and add up to [tex]\(6\)[/tex]. These numbers are [tex]\(9\)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[
y^2 + 9y - 3y - 27
\][/tex]
Grouping the terms and factoring by grouping:
[tex]\[
y(y + 9) - 3(y + 9)
\][/tex]
[tex]\[
(y - 3)(y + 9)
\][/tex]
Substituting [tex]\(y = x^2\)[/tex] back into the factored form:
[tex]\[
(x^2 - 3)(x^2 + 9)
\][/tex]
4. Combine the Factored Forms:
Multiplying our GCF back into the factored form of the quartic polynomial:
[tex]\[
2x(x^2 - 3)(x^2 + 9)
\][/tex]
Therefore, the completely factored form of the polynomial [tex]\(2x^5 + 12x^3 - 54x\)[/tex] is:
[tex]\[
2x(x^2 - 3)(x^2 + 9)
\][/tex]
After comparing to 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]
