Solve 2x^8-2x^12*(-x)

$$2 x ^ { 8 } - 2 x ^ { 12 } \times ( - x )$$

Evaluate

$$2\left(x^{5}+1\right)x^{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 12 and 1 to get 13.

$$2x^{8}-2x^{13}\left(-1\right)$$

Multiply -2 and -1 to get 2.

$$2x^{8}+2x^{13}$$

Factor

$$2\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)x^{8}$$

Solution Steps

Factor out 2.

$$2\left(x^{8}+x^{12}x\right)$$

Consider x^{8}+x^{13}. Factor out x^{8}.

$$x^{8}\left(1+x^{5}\right)$$

Consider 1+x^{5}. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)$$

Rewrite the complete factored expression. Polynomial x^{4}-x^{3}+x^{2}-x+1 is not factored since it does not have any rational roots.

$$2x^{8}\left(x+1\right)\left(x^{4}-x^{3}+x^{2}-x+1\right)$$