Solve f(x)=x^7+x^5+x^3+x

$$f ( x ) = x ^ { 7 } + x ^ { 5 } + x ^ { 3 } + x$$

Factor

$$x\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Solution Steps

Factor out x.

$$x\left(x^{6}+x^{4}+x^{2}+1\right)$$

Consider x^{6}+x^{4}+x^{2}+1. Do the grouping x^{6}+x^{4}+x^{2}+1=\left(x^{6}+x^{4}\right)+\left(x^{2}+1\right), and factor out x^{4} in x^{6}+x^{4}.

$$x^{4}\left(x^{2}+1\right)+x^{2}+1$$

Factor out common term x^{2}+1 by using distributive property.

$$\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}+1,x^{4}+1.

$$x\left(x^{2}+1\right)\left(x^{4}+1\right)$$

Evaluate

$$x^{7}+x^{5}+x^{3}+x$$