SOLUTION: Factoring Polynomials - Please factor the following: w4 - 8w2 - 9

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Question 131936: Factoring Polynomials -
Please factor the following:
w4 - 8w2 - 9
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
I presume you meant w%5E4+-+8w%5E2+-+9

Let x=w%5E2, then substitute:

x%5E2-8x-9

-9%2A1=-9 and -9%2B1=-8, so the factors of x%5E2-8x-9 are:

%28x-9%29%28x%2B1%29

but x=w%5E2, so two of the factors of w%5E4+-+8w%5E2+-+9 are:

%28w%5E2-9%29%28w%5E2%2B1%29

The first of these factors is the difference of two squares and factors directly to:

%28w%5E2-9%29=%28w-3%29%28w%2B3%29

For the other factor, we need to use the fact that a is a zero of a single variable polynomial in x if and only if x-a is a factor of the polynomial.

So, solve w%5E2%2B1=0

w%5E2=-1 => w+=+i or w+=+-i where i%5E2=-1, therefore the two factors of w%5E2%2B1 are %28w-i%29%28w%2Bi%29

In summary:
w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w-i%29%28w%2Bi%29

If you were only supposed to factor over the real numbers, then your answer would be:
w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w%5E2%2B1%29