Solve 1-81x^12

$$1 - 81 x ^ { 12 }$$

Factor

$$\left(3x^{3}-1\right)\left(3x^{3}+1\right)\left(-9x^{6}-1\right)$$

Solution Steps

Rewrite 1-81x^{12} as 1^{2}-\left(9x^{6}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(1-9x^{6}\right)\left(1+9x^{6}\right)$$

Reorder the terms.

$$\left(-9x^{6}+1\right)\left(9x^{6}+1\right)$$

Consider -9x^{6}+1. Rewrite -9x^{6}+1 as 1^{2}-\left(3x^{3}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Reorder the terms.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -3x^{3}+1,3x^{3}+1,9x^{6}+1.

$$\left(-3x^{3}+1\right)\left(3x^{3}+1\right)\left(9x^{6}+1\right)$$

Evaluate

$$1-81x^{12}$$