I have to write a linear factorization of this: f(x) = x4 + 64x2

Question

loser66 · Answer

hey, this problem is not hard for you for sure

anonymous · Answer

These are my choices:
A) f(x) = x^2(8x + i)(8x - i) 
B) f(x) = x^2(x + 8i)^2 
C) f(x) = x^2(x + 8i)(x - 8i) 
D) f(x) = x^2(8x + i)^2

anonymous · Answer

I think it is C. And I know, right @Loser66 ? I feel so dumb. I hate online school.

mathstudent55 · Answer

Start by factoring out a common factor out of x^4 + 64x^2

anonymous · Answer

At first I thought it was b, but then i factored out that answer choice, and it didn't work. Maybe I screwed up...

mathstudent55 · Answer

$f(x) = x4 + 64x2 $

$f(x) = x^2(x^2 + 64)$

$f(x) = x^2(x^2 - (-64))$

$f(x) = x^2(x + \sqrt{-64})(x - \sqrt{-64})$

$f(x) = x^2(x + 8i)(x - 8i)$

anonymous · Answer

Oh, I was right. :p But I didn't figure it out that way. How did you know to do the square root over the -64's, or how would you have if that hadn't been the only choice, seeing that 8i is in every answer choice.

mathstudent55 · Answer

Now they are all linear factors:
$f(x) = x \cdot x(x + 8i)(x - 8i)$

mathstudent55 · Answer

If the last factor after factoring the x^2 term were x^2 - 64, that's the difference of 2 squares which factors into a the product of a sum and a difference.
Instead, we had x^2 + 64.
We can write an addition as a subtraction of the opposite, so x^2 + 64 = x^2 - (-64).

anonymous · Answer

Okay. I got this one. I have two more. Lol.

mathstudent55 · Answer

Sorry, gtg.

anonymous · Answer

Okay.. :( God bless, and thanks for all your help.

mathstudent55 · Answer

You're welcome.