How do you factor x^5+x^3+x^2+1?

1 Answer
Alan P.
Apr 17, 2015

Since x^n is negative for odd values of n
an obvious "solution" (if we pretend this expression =0)
is x=-1
Therefore one of the factors is (x+1)

Using synthetic division
(x^5+x^3+x^2+1)div (x+1)
yields the factors
(x+1)(x^4-x^3+x-1)

Again for (x^4-x^3+x-1) = 0
we have an obvious solution, x=1
So (x-1) is a factor

Using synthetic division again
(x^4-x^3+x-1)div(x-1)
gives the factors
x^4-x^3+x-1) = (x-1)(x^3+1)

Once again
(x^3+1)=0
has an obvious solution
x=-1
which leads to factors
(x^3-1) = (x+1)(x^2-x+1)

The complete factorization is
x^5+x^3+x^2+1

= (x+1)^2(x-1)(x^2-x+1)

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