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 the GCF

The greatest common factor (GCF) of \(2x^{5}\), \(12x^{3}\), and \(-54x\) is \(2x\). Factoring out \(2x\) from the polynomial:
\(2x^{5}+12x^{3}-54x = 2x(x^{4}+6x^{2}-27)\)

Step2: Factor the quadratic in \(x^{2}\)

Let \(y = x^{2}\), then the quadratic \(x^{4}+6x^{2}-27\) becomes \(y^{2}+6y - 27\). We factor this quadratic:
We need two numbers that multiply to \(-27\) and add to \(6\). The numbers are \(9\) and \(-3\). So,
\(y^{2}+6y - 27=(y + 9)(y - 3)\)
Substituting back \(y = x^{2}\), we get:
\(x^{4}+6x^{2}-27=(x^{2}+9)(x^{2}-3)\)

Step3: Factor the difference of squares

The term \(x^{2}-3\) is a difference of squares? Wait, no, \(x^{2}-3=(x+\sqrt{3})(x - \sqrt{3})\)? Wait, no, wait, actually, in the original problem, maybe I made a mistake. Wait, let's re - factor \(x^{4}+6x^{2}-27\) correctly. Wait, no, let's go back. Wait, when we have \(x^{4}+6x^{2}-27\), let's try another way. Wait, maybe I messed up the middle term. Wait, the original polynomial after factoring out \(2x\) is \(x^{4}+6x^{2}-27\). Wait, actually, the correct factoring of \(x^{4}+6x^{2}-27\) is:
We can rewrite \(x^{4}+6x^{2}-27\) as \(x^{4}+9x^{2}-3x^{2}-27\) (splitting the middle term \(6x^{2}\) into \(9x^{2}-3x^{2}\))
Then, group the terms:
\((x^{4}+9x^{2})+(-3x^{2}-27)=x^{2}(x^{2}+9)-3(x^{2}+9)=(x^{2}+9)(x^{2}-3)\)
But \(x^{2}-3\) can be factored as \((x + \sqrt{3})(x-\sqrt{3})\), but that's not in the options. Wait, maybe I made a mistake in the initial factoring. Wait, let's check the original polynomial again. The original polynomial is \(2x^{5}+12x^{3}-54x\). Let's factor \(x^{4}+6x^{2}-27\) again. Wait, maybe the quadratic is \(x^{4}+6x^{2}-27\), but if we consider \(x^{4}+6x^{2}-27\) as \(x^{4}-9x^{2}+15x^{2}-27\)? No, that's not right. Wait, maybe the correct factoring is:
Wait, let's start over. The polynomial is \(2x^{5}+12x^{3}-54x\). Factor out \(2x\): \(2x(x^{4}+6x^{2}-27)\). Now, let's factor \(x^{4}+6x^{2}-27\) as a quadratic in \(x^{2}\). Let \(u=x^{2}\), so \(u^{2}+6u - 27\). Factoring: \(u^{2}+6u - 27=(u + 9)(u - 3)=(x^{2}+9)(x^{2}-3)\). But \(x^{2}-3=(x+\sqrt{3})(x - \sqrt{3})\), but that's not in the options. Wait, maybe there is a mistake in my approach. Wait, let's check the options. Option C is \(2x(x^{2}+3)(x + 3)(x - 3)\). Let's expand option C and see if it matches the original polynomial.

First, expand \((x + 3)(x - 3)=x^{2}-9\). Then, \((x^{2}+3)(x^{2}-9)=x^{4}-9x^{2}+3x^{2}-27=x^{4}-6x^{2}-27\). Wait, that's not the same as \(x^{4}+6x^{2}-27\). Wait, maybe I made a mistake in the sign when factoring. Wait, the original polynomial after factoring out \(2x\) is \(x^{4}+6x^{2}-27\). Let's try to factor it as \(x^{4}-9x^{2}+15x^{2}-27\)? No, that's not helpful. Wait, maybe the correct factoring is:

Wait, let's look at option C: \(2x(x^{2}+3)(x + 3)(x - 3)\). Let's expand it step by step.

First, \((x + 3)(x - 3)=x^{2}-9\). Then, \((x^{2}+3)(x^{2}-9)=x^{4}-9x^{2}+3x^{2}-27=x^{4}-6x^{2}-27\). Then, multiply by \(2x\): \(2x(x^{4}-6x^{2}-27)=2x^{5}-12x^{3}-54x\), which is not the original polynomial. Wait, that's a problem. Wait, maybe I messed up the sign in the middle term. Wait, the original polynomial is \(2x^{5}+12x^{3}-54x\). Let's check option C again. Wait, maybe I made a mistake in the factoring of the quadratic. Let's try another way. Let's take the original polynomial \(2x^{5}+12x^{3}-54x\), factor out \(2x\) to get \(2x(x^{4}+6x^{2}-27)\). Now, let's factor \(x^{4}+6x^{2}-27\) as follows:

We can write \(x^{4}+6x^{2}-27=(x^{2}+a)(x^{2}+b)=x^{4}+(a + b)x^{2}+ab\). So we need \(a + b = 6\) and \(ab=-27\). Solving, we get \(a = 9\) and \(b=-3\). So \(x^{4}+6x^{2}-27=(x^{2}+9)(x^{2}-3)\). Now, \(x^{2}-3=(x+\sqrt{3})(x - \sqrt{3})\), but that's not in the options. Wait, maybe the original polynomial was supposed to be \(2x^{5}-12x^{3}-54x\)? No, the problem says \(2x^{5}+12x^{3}-54x\). Wait, let's check option C again. Wait, maybe there is a typo in my expansion. Wait, let's expand option C:

\(2x(x^{2}+3)(x + 3)(x - 3)=2x(x^{2}+3)(x^{2}-9)=2x(x^{4}-9x^{2}+3x^{2}-27)=2x(x^{4}-6x^{2}-27)=2x^{5}-12x^{3}-54x\). But the original polynomial is \(2x^{5}+12x^{3}-54x\). So there is a sign difference. Wait, maybe the quadratic should be \(x^{4}-6x^{2}-27\) for option C to be correct, but the original polynomial has \(+12x^{3}\). Wait, maybe I made a mistake in factoring out the GCF. Wait, \(2x^{5}+12x^{3}-54x\), factoring out \(2x\) gives \(2x(x^{4}+6x^{2}-27)\). Now, let's factor \(x^{4}+6x^{2}-27\) as \(x^{4}+9x^{2}-3x^{2}-27=x^{2}(x^{2}+9)-3(x^{2}+9)=(x^{2}+9)(x^{2}-3)\). Now, \(x^{2}-3\) can be written as \(x^{2}-(\sqrt{3})^{2}=(x + \sqrt{3})(x - \sqrt{3})\), but that's not in the options. Wait, maybe the problem has a typo, or I am missing something. Wait, let's check option C again. Wait, if we consider that maybe the quadratic was \(x^{4}-6x^{2}-27\), then option C would be correct, but the original polynomial has \(+6x^{2}\). Wait, maybe I made a mistake in the sign when factoring. Wait, let's check the original polynomial again: \(2x^{5}+12x^{3}-54x\). Let's factor out \(2x\): \(2x(x^{4}+6x^{2}-27)\). Now, let's try to factor \(x^{4}+6x^{2}-27\) as \((x^{2}+3)(x^{2}+3)-\) no, that's not helpful. Wait, maybe the correct answer is C, and I made a mistake in the expansion. Let's expand option C again:

\(2x(x^{2}+3)(x + 3)(x - 3)\)

First, \((x + 3)(x - 3)=x^{2}-9\)

Then, \((x^{2}+3)(x^{2}-9)=x^{4}-9x^{2}+3x^{2}-27=x^{4}-6x^{2}-27\)

Then, multiply by \(2x\): \(2x(x^{4}-6x^{2}-27)=2x^{5}-12x^{3}-54x\). But the original polynomial is \(2x^{5}+12x^{3}-54x\). There is a sign difference in the \(x^{3}\) term. Wait, maybe the original polynomial was \(2x^{5}-12x^{3}-54x\), but the problem says \(+12x^{3}\). Wait, maybe there is a mistake in my factoring. Wait, let's check option D: \(2x(x^{2}-3)(x^{2}+9)\). Expand it: \(2x(x^{4}+9x^{2}-3x^{2}-27)=2x(x^{4}+6x^{2}-27)=2x^{5}+12x^{3}-54x\). Oh! Wait, I see my mistake. When I factored \(x^{4}+6x^{2}-27\) as \((x^{2}+9)(x^{2}-3)\), that's correct. And option D is \(2x(x^{2}-3)(x^{2}+9)\), which is the same as \(2x(x^{2}+9)(x^{2}-3)\). Wait, but earlier when I thought about option C, I made a mistake. Wait, no, option C is \(2x(x^{2}+3)(x + 3)(x - 3)\). Wait, let's re - evaluate.

Wait, the polynomial after factoring out \(2x\) is \(2x(x^{4}+6x^{2}-27)\). Now, \(x^{4}+6x^{2}-27=(x^{2})^{2}+6x^{2}-27\). Let's factor this quadratic in \(x^{2}\):

We need two numbers that multiply to \(-27\) and add to \(6\). The numbers are \(9\) and \(-3\). So, \(x^{4}+6x^{2}-27=(x^{2}+9)(x^{2}-3)\). So the factored form is \(2x(x^{2}+9)(x^{2}-3)\), which is option D? Wait, no, option D is \(2x(x^{2}-3)(x^{2}+9)\), which is the same as \(2x(x^{2}+9)(x^{2}-3)\). But wait, let's check the original polynomial. Wait, when we expand option D:

\(2x(x^{2}-3)(x^{2}+9)=2x[(x^{2})(x^{2})+9x^{2}-3x^{2}-27]=2x(x^{4}+6x^{2}-27)=2x^{5}+12x^{3}-54x\), which matches the original polynomial. Wait, but earlier I thought option C was the answer, but I made a mistake in the expansion. Wait, no, option C's expansion gives \(2x^{5}-12x^{3}-54x\), which is not the same as the original polynomial. So the correct answer should be D? Wait, no, wait, 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 \(2x\): \(2x(x^{4}+6x^{2}-27)\)
2. Factor \(x^{4}+6x^{2}-27\) as a quadratic in \(x^{2}\): Let \(u = x^{2}\), then \(u^{2}+6u - 27=(u + 9)(u - 3)=(x^{2}+9)(x^{2}-3)\)
3. So the completely factored form is \(2x(x^{2}+9)(x^{2}-3)=2x(x^{2}-3)(x^{2}+9)\), which is option D.

Wait, but earlier I was confused with option C. Let's confirm by expanding option D:

\(2x(x^{2}-3)(x^{2}+9)=2x[(x^{2})(x^{2})+9x^{2}-3x^{2}-27]=2x(x^{4}+6x^{2}-27)=2x^{5}+12x^{3}-54x\), which is the original polynomial. So the correct answer is D? Wait, but let's check the options again. Wait, option D is \(2x(x^{2}-3)(x^{2}+9)\), and when we factor \(x^{2}-3\), it's a difference of squares? No, \(x^{2}-3=(x+\sqrt{3})(x - \sqrt{3})\), but in the context of the problem, maybe we are supposed to factor it as much as possible over the integers. Wait, \(x^{2}+9\) doesn't factor over the integers, and \(x^{2}-3\) doesn't factor over the integers either. But let's check the options again. Wait, maybe I made a mistake in the initial factoring. Wait, let's try factoring \(x^{4}+6x^{2}-27\) as \((x^{2}+3)(x^{2}-9)+12x^{2}\)? No, that's not helpful. Wait, no, the correct factoring of \(x^{4}+6x^{2}-27\) is \((x^{2}+9)(x^{2}-3)\), so the factored form is \(2x(x^{2}+9)(x^{2}-3)\), which is option D.

Wait, but let's check option C again. Option C is \(2x(x^{2}+3)(x + 3)(x - 3)\). Let's expand it:

\((x + 3)(x - 3)=x^{2}-9\)

\((x^{2}+3)(x^{2}-9)=x^{4}-9x^{2}+3x^{2}-27=x^{4}-6x^{2}-27\)

\(2x(x^{4}-6x^{2}-27)=2x^{5}-12x^{3}-54x\), which is not the original polynomial. So the correct answer is D. Wait, but the original polynomial has \(+12x^{3}\), and when we expand option D, we get \(+12x^{3}\), which matches. So the correct answer is D.

Answer:

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