Solve -11x^3+77x^2-132x

$$- 11 x ^ { 3 } + 77 x ^ { 2 } - 132 x$$

Factor

$$11x\left(3-x\right)\left(x-4\right)$$

Solution Steps

Factor out 11.

$$11\left(-x^{3}+7x^{2}-12x\right)$$

Consider -x^{3}+7x^{2}-12x. Factor out x.

$$x\left(-x^{2}+7x-12\right)$$

Consider -x^{2}+7x-12. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=4 b=3$$

Rewrite -x^{2}+7x-12 as \left(-x^{2}+4x\right)+\left(3x-12\right).

$$\left(-x^{2}+4x\right)+\left(3x-12\right)$$

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

$$-x\left(x-4\right)+3\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(-x+3\right)$$

Rewrite the complete factored expression.

$$11x\left(x-4\right)\left(-x+3\right)$$

Evaluate

$$11x\left(3-x\right)\left(x-4\right)$$