Solve x^3+9x^2-25x-33

$${ x }^{ 3 } +9 { x }^{ 2 } -25x-33$$

Factor

$$\left(x-3\right)\left(x+1\right)\left(x+11\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -33 and q divides the leading coefficient 1. One such root is -11. Factor the polynomial by dividing it by x+11.

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

Consider x^{2}-2x-3. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-3\right)=-3$$

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. The only such pair is the system solution.

$$a=-3 b=1$$

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

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

Factor out x in x^{2}-3x.

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

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$\left(x-3\right)\left(x+1\right)\left(x+11\right)$$

Evaluate

$$\left(x-3\right)\left(x+1\right)\left(x+11\right)$$