Solve 4x^3-66x^2+216x

$$4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$

Factor

$$2x\left(x-12\right)\left(2x-9\right)$$

Solution Steps

Factor out 2.

$$2\left(2x^{3}-33x^{2}+108x\right)$$

Consider 2x^{3}-33x^{2}+108x. Factor out x.

$$x\left(2x^{2}-33x+108\right)$$

Consider 2x^{2}-33x+108. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+108. To find a and b, set up a system to be solved.

$$a+b=-33 ab=2\times 108=216$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 216.

$$-1,-216 -2,-108 -3,-72 -4,-54 -6,-36 -8,-27 -9,-24 -12,-18$$

Calculate the sum for each pair.

$$-1-216=-217 -2-108=-110 -3-72=-75 -4-54=-58 -6-36=-42 -8-27=-35 -9-24=-33 -12-18=-30$$

The solution is the pair that gives sum -33.

$$a=-24 b=-9$$

Rewrite 2x^{2}-33x+108 as \left(2x^{2}-24x\right)+\left(-9x+108\right).

$$\left(2x^{2}-24x\right)+\left(-9x+108\right)$$

Factor out 2x in the first and -9 in the second group.

$$2x\left(x-12\right)-9\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(2x-9\right)$$

Rewrite the complete factored expression.

$$2x\left(x-12\right)\left(2x-9\right)$$

Evaluate

$$2x\left(x-12\right)\left(2x-9\right)$$