Solve x^4-2x^3+8x-16

$${ x }^{ 4 } -2 { x }^{ 3 } +8x-16$$

Factor

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

Solution Steps

Do the grouping x^{4}-2x^{3}+8x-16=\left(x^{4}-2x^{3}\right)+\left(8x-16\right), and factor out x^{3} in the first and 8 in the second group.

$$x^{3}\left(x-2\right)+8\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x^{3}+8\right)$$

Consider x^{3}+8. Rewrite x^{3}+8 as x^{3}+2^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

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

Rewrite the complete factored expression. Polynomial x^{2}-2x+4 is not factored since it does not have any rational roots.

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

Evaluate

$$x^{4}-2x^{3}+8x-16$$