Solve f(x)=x^4+4x^3-16x^2-64x

$$f ( x ) = x ^ { 4 } + 4 x ^ { 3 } - 16 x ^ { 2 } - 64 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(x^{3}+4x^{2}-16x-64\right)$$

Consider x^{3}+4x^{2}-16x-64. Do the grouping x^{3}+4x^{2}-16x-64=\left(x^{3}+4x^{2}\right)+\left(-16x-64\right), and factor out x^{2} in the first and -16 in the second group.

$$x^{2}\left(x+4\right)-16\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x^{2}-16\right)$$

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

$$\left(x-4\right)\left(x+4\right)$$

Rewrite the complete factored expression.

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

Evaluate

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