Solve x^10-5x^5+4

$$x ^ { 10 } - 5 x ^ { 5 } + 4$$

Factor

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

Solution Steps

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{10} and m divides the constant factor 4. One such factor is x^{5}-4. Factor the polynomial by dividing it by this factor.

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

Consider x^{5}-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.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{5}-4,x^{4}+x^{3}+x^{2}+x+1.

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

Evaluate

$$x^{10}-5x^{5}+4$$