Solve 4x^7-8x^8+4x^4

$$4 { x }^{ 7 } -8 { x }^{ 8 } +4 { x }^{ 4 }$$

Factor

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

Solution Steps

Factor out 4.

$$4\left(x^{7}-2x^{8}+x^{4}\right)$$

Consider x^{7}-2x^{8}+x^{4}. Factor out x^{4}.

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

Consider x^{3}-2x^{4}+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 -2. One such root is 1. Factor the polynomial by dividing it by x-1.

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

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

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

Evaluate

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