Solve 32-2x^16

$$32 - 2 x ^ { 16 }$$

Factor

$$2\left(x^{4}-2\right)\left(x^{4}+2\right)\left(-x^{8}-4\right)$$

Solution Steps

Factor out 2.

$$2\left(16-x^{16}\right)$$

Consider 16-x^{16}. Rewrite 16-x^{16} as 4^{2}-\left(-x^{8}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(4+x^{8}\right)\left(4-x^{8}\right)$$

Reorder the terms.

$$\left(x^{8}+4\right)\left(-x^{8}+4\right)$$

Consider -x^{8}+4. Rewrite -x^{8}+4 as 2^{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).

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

Reorder the terms.

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -x^{4}+2,x^{4}+2,x^{8}+4.

$$2\left(x^{8}+4\right)\left(x^{4}+2\right)\left(-x^{4}+2\right)$$

Evaluate

$$32-2x^{16}$$