Solve 7x^3-36x^2-36x

$$7 x ^ { 3 } - 36 x ^ { 2 } - 36 x$$

Factor

$$x\left(x-6\right)\left(7x+6\right)$$

Solution Steps

Factor out x.

$$x\left(7x^{2}-36x-36\right)$$

Consider 7x^{2}-36x-36. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-36 ab=7\left(-36\right)=-252$$

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 -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Calculate the sum for each pair.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

The solution is the pair that gives sum -36.

$$a=-42 b=6$$

Rewrite 7x^{2}-36x-36 as \left(7x^{2}-42x\right)+\left(6x-36\right).

$$\left(7x^{2}-42x\right)+\left(6x-36\right)$$

Factor out 7x in the first and 6 in the second group.

$$7x\left(x-6\right)+6\left(x-6\right)$$

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

$$\left(x-6\right)\left(7x+6\right)$$

Rewrite the complete factored expression.

$$x\left(x-6\right)\left(7x+6\right)$$

Evaluate

$$x\left(x-6\right)\left(7x+6\right)$$