Solve n^5-5n^3+4n

$$n ^ { 5 } - 5 n ^ { 3 } + 4 n$$

Factor

$$n\left(n-2\right)\left(n-1\right)\left(n+1\right)\left(n+2\right)$$

Solution Steps

Factor out n.

$$n\left(n^{4}-5n^{2}+4\right)$$

Consider n^{4}-5n^{2}+4. Find one factor of the form n^{k}+m, where n^{k} divides the monomial with the highest power n^{4} and m divides the constant factor 4. One such factor is n^{2}-4. Factor the polynomial by dividing it by this factor.

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

Consider n^{2}-4. Rewrite n^{2}-4 as n^{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).

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

Consider n^{2}-1. Rewrite n^{2}-1 as n^{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).

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

Rewrite the complete factored expression.

$$n\left(n-2\right)\left(n+2\right)\left(n-1\right)\left(n+1\right)$$

Evaluate

$$n\left(n^{4}-5n^{2}+4\right)$$