Solve x^5-17x^3+16x

$$x ^ { 5 } - 17 x ^ { 3 } + 16 x$$

Factor

$$x\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)$$

Solution Steps

Factor out x.

$$x\left(x^{4}-17x^{2}+16\right)$$

Consider x^{4}-17x^{2}+16. Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor 16. One such factor is x^{2}-16. Factor the polynomial by dividing it by this factor.

$$\left(x^{2}-16\right)\left(x^{2}-1\right)$$

Consider x^{2}-16. Rewrite x^{2}-16 as x^{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).

$$\left(x-4\right)\left(x+4\right)$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$x\left(x-4\right)\left(x+4\right)\left(x-1\right)\left(x+1\right)$$

Evaluate

$$x\left(x^{4}-17x^{2}+16\right)$$