Solve -a^4+81

$$- a ^ { 4 } + 81$$

Factor

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

Solution Steps

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

$$\left(9+a^{2}\right)\left(9-a^{2}\right)$$

Reorder the terms.

$$\left(a^{2}+9\right)\left(-a^{2}+9\right)$$

Consider -a^{2}+9. Rewrite -a^{2}+9 as 3^{2}-a^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(3-a\right)\left(3+a\right)$$

Reorder the terms.

$$\left(-a+3\right)\left(a+3\right)$$

Rewrite the complete factored expression. Polynomial a^{2}+9 is not factored since it does not have any rational roots.

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

Evaluate

$$81-a^{4}$$