Solve 8x^3-6x^2-2x

$$8 { x }^{ 3 } -6 { x }^{ 2 } -2x$$

Factor

$$2x\left(x-1\right)\left(4x+1\right)$$

Solution Steps

Factor out 2.

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

Consider 4x^{3}-3x^{2}-x. Factor out x.

$$x\left(4x^{2}-3x-1\right)$$

Consider 4x^{2}-3x-1. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-1\right)=-4$$

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 -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

Rewrite 4x^{2}-3x-1 as \left(4x^{2}-4x\right)+\left(x-1\right).

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

Factor out 4x in 4x^{2}-4x.

$$4x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(4x+1\right)$$

Rewrite the complete factored expression.

$$2x\left(x-1\right)\left(4x+1\right)$$

Evaluate

$$2x\left(x-1\right)\left(4x+1\right)$$