Solve x^8-15x^4-16

$${ x }^{ 8 } -15 { x }^{ 4 } -16$$

Factor

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

Solution Steps

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{8} and m divides the constant factor -16. One such factor is x^{4}-16. Factor the polynomial by dividing it by this factor.

$$\left(x^{4}-16\right)\left(x^{4}+1\right)$$

Consider x^{4}-16. Rewrite x^{4}-16 as \left(x^{2}\right)^{2}-4^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Consider x^{2}-4. Rewrite x^{2}-4 as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)$$

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

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

Evaluate

$$x^{8}-15x^{4}-16$$