Solve x^2+x^2x^12

$$x ^ { 2 } + x ^ { 2 } x ^ { 12 }$$

Factor

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

Solution Steps

Factor out x^{2}.

$$x^{2}\left(1+x^{12}\right)$$

Consider 1+x^{12}. Rewrite 1+x^{12} as \left(x^{4}\right)^{3}+1^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{8}-x^{4}+1,x^{4}+1.

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

Evaluate

$$x^{14}+x^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 12 to get 14.

$$x^{2}+x^{14}$$