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Need this factored x^5-2x^4+2x

I began by factoring out x^2(x^3-2x^2+2), but need the second polynomial factored down now. Will be 5 total roots (of which, + or - zero I think are two of them) Imaginary numbers are OK, just use i. Please use fractions only, no decimals!

Answer
Hi Dakota, I will be happy to help.

Now, you said you factored out a x^2.  As shown here, the last term is 2x, not 2x^2 so you can at most factor an x.  

First, factor out the x to get

x(x^4 - 2x^3 + 2)

In order to determine if x^4 - 2x^3 + 2 can be factored, determine if any of the factors, both positive and negative, of 2 are roots by plugging them into the expression and evaluating to see if any come to equal 0.

The only factors of 2 are: -2, -1, 1, 2
Using a calculator, you will find, after plugging these values in, none of the factors are roots.

Therefore the expression is factored completely
x(x^4 - 2x^3 + 2)

I will also look at factoring x*5 - 2x^4 + 2x^2 just in case there was a typo.
Factoring out the x^2 gets

x^2(x^3 - 2x^2 + 2)

Again, the factors of 2 are: -2, -1, 1, 2
However, using these values, none are a root, so the expression is factored completely.

I'm glad I can be of assistance.
Please let me know if you have any questions.

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