Solve x^2012-x^2016=

$$x ^ { 2012 } - x ^ { 2016 } =$$

Factor

$$\left(x-1\right)\left(x+1\right)\left(-x^{2}-1\right)x^{2012}$$

Solution Steps

Factor out x^{2012}.

$$x^{2012}\left(1-x^{4}\right)$$

Consider 1-x^{4}. Rewrite 1-x^{4} as 1^{2}-\left(-x^{2}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Reorder the terms.

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

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

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

Reorder the terms.

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

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

$$x^{2012}\left(x^{2}+1\right)\left(-x+1\right)\left(x+1\right)$$

Evaluate

$$x^{2012}-x^{2016}$$