Solve 36x^3-30x^2-36x

$$36 x ^ { 3 } - 30 x ^ { 2 } - 36 x$$

Factor

$$6x\left(2x-3\right)\left(3x+2\right)$$

Solution Steps

Factor out 6.

$$6\left(6x^{3}-5x^{2}-6x\right)$$

Consider 6x^{3}-5x^{2}-6x. Factor out x.

$$x\left(6x^{2}-5x-6\right)$$

Consider 6x^{2}-5x-6. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-6\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

Rewrite 6x^{2}-5x-6 as \left(6x^{2}-9x\right)+\left(4x-6\right).

$$\left(6x^{2}-9x\right)+\left(4x-6\right)$$

Factor out 3x in the first and 2 in the second group.

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

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

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

Rewrite the complete factored expression.

$$6x\left(2x-3\right)\left(3x+2\right)$$

Evaluate

$$6x\left(2x-3\right)\left(3x+2\right)$$