Solve by Factoring x^3+2x^2-5x-6=0

Problem

x3+2*x2−5*x−6=0

Solution

1. Identify a possible integer root using the Rational Root Theorem. Testing x=−1

(−1)3+2*(−1)2−5*(−1)−6=−1+2+5−6=0

Since x=−1 is a root, (x+1) is a factor.

2. Divide the polynomial by the known factor (x+1) using synthetic division or long division to find the remaining quadratic factor:

(x3+2*x2−5*x−6)÷(x+1)=x2+x−6

3. Factor the resulting quadratic expression x2+x−6 by finding two numbers that multiply to −6 and add to 1

x2+x−6=(x+3)*(x−2)

4. Rewrite the original equation in its fully factored form:

(x+1)*(x+3)*(x−2)=0

5. Solve for x by setting each linear factor equal to zero:

x+1=0⇒x=−1

x+3=0⇒x=−3

x−2=0⇒x=2

Final Answer

x={−3,−1,2}

Want more problems? Check here!