SOLUTION: write the polynomial function f(x)= x^5+x^3+2x^2-12x+8 as the product of linear factors

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Question 835252: write the polynomial function
f(x)= x^5+x^3+2x^2-12x+8
as the product of linear factors

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=+x%5E5%2Bx%5E3%2B2x%5E2-12x%2B8=%28x%2B2%29%28x-1%29%5E2%28x%5E2%2B2%29
x%5E2%2B2 is not a linear factor.
It is a quadratic polynomial, and can only be factored using imaginary numbers.

HOW DO WE GET THAT?
Noticing that f%281%29=1%5E5%2B1%5E3%2B2%2A1%5E2-12%2A1%2B8=1%2B1%2B2-12%2B8=0 ,
we realize that x-1 is a factor of f%28x%29 .
Dividing, we find that

Since g%281%29=1%5E4%2B1%5E3%2B2%2A1%5E2%2B4%2A1-8=1%2B1%2B2%2B4-8=0 ,
x-1 must be a factor of g%28x%29 .
Dividing, we find that
g%28x%29%2F%28x-1%29=%28x%5E4%2Bx%5E3%2B2x%5E2%2B4x-8%29%2F%28x-1%29=x%5E3%2B2x%5E2%2B4x%2B8 .
So g%28x%29=%28x-1%29%28x%5E3%2B2x%5E2%2B4x%2B8%29 and f%28x%29=%28x-1%29g%28x%29 ,
which means that


At this point, either we may realize that x%5E3%2B2x%5E2%2B4x%2B8 is zero for x=-2 ,
or we may realize that x%5E3%2B2x%5E2%2B4x%2B8 is divisible by x%2B2 .
In either case we divide and find that x%5E3%2B2x%5E2%2B4x%2B8=%28x%2B2%29%28x%5E2%2B4%29 .
So, putting it all together,
f%28x%29=%28x-1%29%5E2%28x%5E3%2B2x%5E2%2B4x%2B8%29=%28x-1%29%5E2%28x%2B2%29%28x%5E2%2B4%29 .

Alternatively, we may realize that

and since %28x%5E3%2B2x%5E2%2B4x%2B8%29%28x-2%29=%28x%5E2%2B2%29%28x%2B2%29%28x-2%29 ,
dividing both sides by %28x-2%29 we conclude that
%28x%5E3%2B2x%5E2%2B4x%2B8%29=%28x%5E2%2B2%29%28x%2B2%29