Solve 1/4x^6-9x^8+81x^10

$$\frac { 1 } { 4 } x ^ { 6 } - 9 x ^ { 8 } + 81 x ^ { 10 }$$

Factor

$$\frac{\left(18x^{2}-1\right)^{2}x^{6}}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{x^{6}-36x^{8}+324x^{10}}{4}$$

Consider x^{6}-36x^{8}+324x^{10}. Factor out x^{6}.

$$x^{6}\left(1-36x^{2}+324x^{4}\right)$$

Consider 1-36x^{2}+324x^{4}. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=18x^{2} and b=1.

$$\left(18x^{2}-1\right)^{2}$$

Rewrite the complete factored expression. Polynomial 18x^{2}-1 is not factored since it does not have any rational roots.

$$\frac{x^{6}\left(18x^{2}-1\right)^{2}}{4}$$

Evaluate

$$\frac{\left(18x^{2}-1\right)^{2}x^{6}}{4}$$