Solve x^7-x^3

$${ x }^{ 7 } - { x }^{ 3 }$$

Factor

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

Solution Steps

Factor out x^{3}.

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

Consider x^{4}-1. Rewrite x^{4}-1 as \left(x^{2}\right)^{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).

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

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. Polynomial x^{2}+1 is not factored since it does not have any rational roots.

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

Evaluate

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