Hey I need some help with this problem

Factor the polynomial x^8-98x^4+1 into two non-constant polynomials with

integral coefficients.

I want to see steps on how to do this problem thanks.

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- Nov 10th 2012, 08:26 PMgfbrdFactoring help
Hey I need some help with this problem

Factor the polynomial x^8-98x^4+1 into two non-constant polynomials with

integral coefficients.

I want to see steps on how to do this problem thanks. - Nov 10th 2012, 08:40 PMProve ItRe: Factoring help
Start by letting $\displaystyle \displaystyle X = x^4$, then the equation becomes $\displaystyle \displaystyle \begin{align*} X^2 - 98X + 1 \end{align*}$. This doesn't factorise over the integers, but does factorise over the real numbers by completing the square.

- Nov 10th 2012, 09:42 PMgfbrdRe: Factoring help
I completed the square and got

(x-49)^2 +2400

what do I do next? - Nov 10th 2012, 09:43 PMProve ItRe: Factoring help
That can't possibly be right - expand it out and check...

- Nov 10th 2012, 09:58 PMgfbrdRe: Factoring help
sorry it should be -2400

(x-49)^2 - 2400 = (x-49)(x-49) - 2400 = x^2 -49x - 49x + 2401 -2400 = x^2 - 98x + 1

ok so when completing the square i get

(x-49)^2 - 2400

what do I do next? - Nov 10th 2012, 10:01 PMProve ItRe: Factoring help
Write it as $\displaystyle \displaystyle \begin{align*} (X - 49)^2 - \left( 20\sqrt{6} \right)^2 \end{align*}$ and then use the Difference of Two Squares rule to factorise it.

- Nov 10th 2012, 10:27 PMgfbrdRe: Factoring help
so it is ((x-49)^2 + (20sqrt(6))^2)((x-49)^2 - (20sqrt(6))^2))

what do I do from here? - Nov 11th 2012, 08:58 AMSorobanRe: Factoring help
Hello, gfbrd!

This takes a little Imagination . . .

Quote:

Factor $\displaystyle x^8-98x^4+1$ into two non-constant polynomials with integral coefficients.

$\displaystyle \begin{array}{cc}\text{We are given:} & x^8 - 98x^4 + 1 \\ \text{Add/subtract } 100x^4\!: & x^8 - 98x^4 {\color{blue}+ 100x^4} + 1 {\color{blue}- 100x^4} \\ \text{And we have:} & x^8 + 2x^4 + 1 - 100x^4 \\ \text{Factor:} & (x^4 +1)^2 - (10x^2)^2 \\ \text{Factor:} & (x^4+1-10x^2)(x^4+1+10x^2) \\ \text{Therefore:} & (x^4 - 10x^2+1)(x^4+10x^2+1) \end{array}$

- Nov 11th 2012, 09:11 AMgfbrdRe: Factoring help
lol sorry my imagination isn't really as good as yours

thanks a lot soroban