Solve x^12*x^2+x^3

$$x ^ { 12 } \times x ^ { 2 } + x ^ { 3 }$$

Factor

$$\left(x+1\right)x^{3}\left(x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)$$

Solution Steps

Factor out x^{3}.

$$x^{3}\left(x^{9}x^{2}+1\right)$$

Consider x^{11}+1. 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.

$$\left(x+1\right)\left(x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)$$

Rewrite the complete factored expression. Polynomial x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1 is not factored since it does not have any rational roots.

$$x^{3}\left(x+1\right)\left(x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\right)$$

Evaluate

$$x^{14}+x^{3}$$

Solution Steps

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

$$x^{14}+x^{3}$$