Solve -9x^3+198x^2+207x

$$- 9 x ^ { 3 } + 198 x ^ { 2 } + 207 x$$

Factor

$$-9x\left(x-23\right)\left(x+1\right)$$

Solution Steps

Factor out 9.

$$9\left(-x^{3}+22x^{2}+23x\right)$$

Consider -x^{3}+22x^{2}+23x. Factor out x.

$$x\left(-x^{2}+22x+23\right)$$

Consider -x^{2}+22x+23. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+23. To find a and b, set up a system to be solved.

$$a+b=22 ab=-23=-23$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=23 b=-1$$

Rewrite -x^{2}+22x+23 as \left(-x^{2}+23x\right)+\left(-x+23\right).

$$\left(-x^{2}+23x\right)+\left(-x+23\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(x-23\right)-\left(x-23\right)$$

Factor out common term x-23 by using distributive property.

$$\left(x-23\right)\left(-x-1\right)$$

Rewrite the complete factored expression.

$$9x\left(x-23\right)\left(-x-1\right)$$

Evaluate

$$-9x\left(x-23\right)\left(x+1\right)$$