Solve 8x^3-16x^2-192x

$$8 x ^ { 3 } - 16 x ^ { 2 } - 192 x$$

Factor

$$8x\left(x-6\right)\left(x+4\right)$$

Solution Steps

Factor out 8.

$$8\left(x^{3}-2x^{2}-24x\right)$$

Consider x^{3}-2x^{2}-24x. Factor out x.

$$x\left(x^{2}-2x-24\right)$$

Consider x^{2}-2x-24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).

$$\left(x^{2}-6x\right)+\left(4x-24\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-6\right)+4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+4\right)$$

Rewrite the complete factored expression.

$$8x\left(x-6\right)\left(x+4\right)$$

Evaluate

$$8x\left(x-6\right)\left(x+4\right)$$