Solve 9x^3-3x^2-2x

$$9 { x }^{ 3 } -3 { x }^{ 2 } -2x$$

Factor

$$x\left(3x-2\right)\left(3x+1\right)$$

Solution Steps

Factor out x.

$$x\left(9x^{2}-3x-2\right)$$

Consider 9x^{2}-3x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=9\left(-2\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -3.

$$a=-6 b=3$$

Rewrite 9x^{2}-3x-2 as \left(9x^{2}-6x\right)+\left(3x-2\right).

$$\left(9x^{2}-6x\right)+\left(3x-2\right)$$

Factor out 3x in 9x^{2}-6x.

$$3x\left(3x-2\right)+3x-2$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(3x+1\right)$$

Rewrite the complete factored expression.

$$x\left(3x-2\right)\left(3x+1\right)$$

Evaluate

$$x\left(3x-2\right)\left(3x+1\right)$$