Solve 3x+2x+2x-x^8-6x^8

$$3 x + 2 x + 2 x - x ^ { 8 } - 6 x ^ { 8 }$$

Evaluate

$$7x-7x^{8}$$

Solution Steps

Combine 3x and 2x to get 5x.

$$5x+2x-x^{8}-6x^{8}$$

Combine 5x and 2x to get 7x.

$$7x-x^{8}-6x^{8}$$

Combine -x^{8} and -6x^{8} to get -7x^{8}.

$$7x-7x^{8}$$

Factor

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

Solution Steps

Factor out x.

$$x\left(7-7x^{7}\right)$$

Consider 3+2+2-x^{7}-6x^{7}. Multiply and combine like terms.

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

Consider -7x^{7}+7. Factor out 7.

$$7\left(-x^{7}+1\right)$$

Consider -x^{7}+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.

$$\left(x-1\right)\left(-x^{6}-x^{5}-x^{4}-x^{3}-x^{2}-x-1\right)$$

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

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