Question

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 the GCF

The given polynomial is \(2x^5 + 12x^3 - 54x\). First, factor out the greatest common factor (GCF), which is \(2x\).
\[ 2x^5 + 12x^3 - 54x = 2x(x^4 + 6x^2 - 27) \]

Step2: Factor the quadratic in terms of \(x^2\)

Let \(y = x^2\), then the quadratic \(x^4 + 6x^2 - 27\) becomes \(y^2 + 6y - 27\). Factor this quadratic:
\[ y^2 + 6y - 27 = (y + 9)(y - 3) \]
Substitute back \(y = x^2\):
\[ (x^2 + 9)(x^2 - 3) \]
Wait, that doesn't seem right. Wait, maybe I made a mistake. Let's try another way. Wait, actually, \(x^4 + 6x^2 - 27\) can be factored as \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\)? No, wait, \(x^2 + 9\) and \(x^2 - 3\) multiply to \(x^4 + 6x^2 - 27\)? Let's check: \((x^2 + 9)(x^2 - 3)=x^4 - 3x^2 + 9x^2 - 27 = x^4 + 6x^2 - 27\). But that's not the correct path. Wait, maybe the original polynomial after factoring out \(2x\) is \(2x(x^4 + 6x^2 - 27)\), but maybe I should factor \(x^4 + 6x^2 - 27\) as a quadratic in \(x^2\) but with a different approach. Wait, no, actually, let's try to factor \(x^4 + 6x^2 - 27\) as \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\) is correct, but that's not one of the options. Wait, maybe I made a mistake in the GCF step. Wait, no, the GCF is \(2x\). Wait, maybe the polynomial is \(2x^5 + 12x^3 - 54x\), so factoring out \(2x\) gives \(2x(x^4 + 6x^2 - 27)\). Wait, but let's check the options. Option C is \(2x(x^2 + 3)(x + 3)(x - 3)\). Let's expand \(x^2 - 9=(x + 3)(x - 3)\), so \(x^4 - 9x^2 + 3x^2 - 27\)? No, wait, let's expand option C:
\[ 2x(x^2 + 3)(x + 3)(x - 3)=2x(x^2 + 3)(x^2 - 9)=2x(x^4 - 9x^2 + 3x^2 - 27)=2x(x^4 - 6x^2 - 27) \]
Wait, that's not the original polynomial. Wait, maybe I messed up the sign. Wait, the original polynomial is \(2x^5 + 12x^3 - 54x\). Let's try factoring \(x^4 + 6x^2 - 27\) as \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\) is incorrect. Wait, no, let's use the quadratic formula for \(y^2 + 6y - 27\). The discriminant is \(36 + 108 = 144\), so roots are \(\frac{-6 \pm 12}{2}\), which are \(3\) and \(-9\). So \(y^2 + 6y - 27=(y + 9)(y - 3)\), so \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\). But that's not matching the options. Wait, maybe the original polynomial was \(2x^5 + 12x^3 - 54x\), but maybe there's a typo, or maybe I made a mistake. Wait, let's check option C: \(2x(x^2 + 3)(x + 3)(x - 3)=2x(x^2 + 3)(x^2 - 9)=2x(x^4 - 9x^2 + 3x^2 - 27)=2x(x^4 - 6x^2 - 27)\). No, that's \(2x^5 - 12x^3 - 54x\), which is different from the original. Wait, the original is \(+12x^3\). Oh! Wait, I see my mistake. The quadratic in \(x^2\) is \(x^4 + 6x^2 - 27\), but if we factor it as \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\), but that's not correct. Wait, no, let's try to factor \(x^4 + 6x^2 - 27\) as \(x^4 + 6x^2 - 27=(x^2 + 9)(x^2 - 3)\) is wrong. Wait, actually, \(x^4 + 6x^2 - 27\) can be written as \(x^4 + 9x^2 - 3x^2 - 27=x^2(x^2 + 9) - 3(x^2 + 9)=(x^2 - 3)(x^2 + 9)\). Ah! So that's option D: \(2x(x^2 - 3)(x^2 + 9)\). Let's expand option D: \(2x(x^2 - 3)(x^2 + 9)=2x(x^4 + 9x^2 - 3x^2 - 27)=2x(x^4 + 6x^2 - 27)=2x^5 + 12x^3 - 54x\), which matches the original polynomial. Wait, but earlier I thought option C, but no, option D is correct? Wait, but let's check the options again. Wait, the original polynomial is \(2x^5 + 12x^3 - 54x\). Let's factor it step by step:

1. Factor out GCF: \(2x(x^4 + 6x^2 - 27)\)
2. Let \(u = x^2\), so \(u^2 + 6u - 27\)
3. Factor \(u^2 + 6u - 27\): find two numbers that multiply to -27 and add to 6. Those numbers are 9 and -3. So \(u^2 + 6u - 27=(u + 9)(u - 3)\)
4. Substitute back \(u = x^2\): \((x^2 + 9)(x^2 - 3)\)
5. So the fully factored form is \(2x(x^2 + 9)(x^2 - 3)\), which is option D. Wait, but earlier I thought option D was \(2x(x^2 - 3)(x^2 + 9)\), which is the same as \(2x(x^2 + 9)(x^2 - 3)\). So that's correct. Wait, but let's check the options again. Option D is \(2x(x^2 - 3)(x^2 + 9)\), which when expanded is \(2x(x^4 + 9x^2 - 3x^2 - 27)=2x(x^4 + 6x^2 - 27)=2x^5 + 12x^3 - 54x\), which matches the original polynomial. So the correct answer is D? Wait, but let me check again. Wait, maybe I made a mistake in the initial factoring. Wait, the original polynomial is \(2x^5 + 12x^3 - 54x\). Let's factor out \(2x\): \(2x(x^4 + 6x^2 - 27)\). Then factor \(x^4 + 6x^2 - 27\) as \((x^2 + 9)(x^2 - 3)\), so the fully factored form is \(2x(x^2 + 9)(x^2 - 3)\), which is option D. So the correct answer is D.

Wait, but let me check option C again. Option C is \(2x(x^2 + 3)(x + 3)(x - 3)=2x(x^2 + 3)(x^2 - 9)=2x(x^4 - 9x^2 + 3x^2 - 27)=2x(x^4 - 6x^2 - 27)=2x^5 - 12x^3 - 54x\), which is different from the original (the original has +12x^3). So option C is incorrect. Option D is correct.

Answer:

D. \(2x(x^2 - 3)(x^2 + 9)\)