Solve -x^6x-x

$$- x ^ { 6 } x - x$$

Factor

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

Solution Steps

Factor out x.

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

Consider -x^{6}-1. Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power -x^{6} and n divides the constant factor -1. One such factor is x^{2}+1. Factor the polynomial by dividing it by this factor.

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

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

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

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$-x^{7}-x$$