Solve -x^8+81

$$- x ^ { 8 } + 81$$

Factor

$$\left(x^{2}-3\right)\left(x^{2}+3\right)\left(-x^{4}-9\right)$$

Solution Steps

Rewrite -x^{8}+81 as 9^{2}-\left(-x^{4}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(9+x^{4}\right)\left(9-x^{4}\right)$$

Reorder the terms.

$$\left(x^{4}+9\right)\left(-x^{4}+9\right)$$

Consider -x^{4}+9. Rewrite -x^{4}+9 as 3^{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).

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

Reorder the terms.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -x^{2}+3,x^{2}+3,x^{4}+9.

$$\left(-x^{2}+3\right)\left(x^{2}+3\right)\left(x^{4}+9\right)$$

Evaluate

$$81-x^{8}$$