Solve x^3-9x^2-22x

$$x ^ { 3 } - 9 x ^ { 2 } - 22 x$$

Factor

$$x\left(x-11\right)\left(x+2\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}-9x-22\right)$$

Consider x^{2}-9x-22. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-22. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\left(-22\right)=-22$$

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 -22.

$$1,-22 2,-11$$

Calculate the sum for each pair.

$$1-22=-21 2-11=-9$$

The solution is the pair that gives sum -9.

$$a=-11 b=2$$

Rewrite x^{2}-9x-22 as \left(x^{2}-11x\right)+\left(2x-22\right).

$$\left(x^{2}-11x\right)+\left(2x-22\right)$$

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

$$x\left(x-11\right)+2\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$x\left(x-11\right)\left(x+2\right)$$

Evaluate

$$x\left(x-11\right)\left(x+2\right)$$