Solve x^2+x^4+x^6-x^10

$$x ^ { 2 } + x ^ { 4 } + x ^ { 6 } - x ^ { 10 }$$

Factor

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

Solution Steps

Factor out x^{2}.

$$x^{2}\left(1+x^{2}+x^{4}-x^{8}\right)$$

Consider 1+x^{2}+x^{4}-x^{8}. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power -x^{8} and n divides the constant factor 1. One such factor is x^{2}+1. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}+1,-x^{6}+x^{4}+1.

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

Evaluate

$$x^{2}+x^{4}+x^{6}-x^{10}$$