Find the Roots (Zeros) f(x)=x^3+2x^2-5x-6

Problem

ƒ(x)=x3+2*x2−5*x−6

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term −6 divided by factors of the leading coefficient 1 The possible roots are ±1,±2,±3,±6

2. Test the value x=−1 by substituting it into the function: ƒ*(−1)=(−1)3+2*(−1)2−5*(−1)−6=−1+2+5−6=0 Since ƒ*(−1)=0 (x+1) is a factor.

3. Divide the polynomial x3+2*x2−5*x−6 by (x+1) using synthetic division or long division to find the remaining quadratic factor.

4. Perform the division:

(x3+2*x2−5*x−6)/(x+1)=x2+x−6

5. Factor the resulting quadratic expression x2+x−6 by finding two numbers that multiply to −6 and add to 1 These numbers are 3 and −2

x2+x−6=(x+3)*(x−2)

6. Set each factor to zero to find all roots of the function:

x+1=0⇒x=−1

x+3=0⇒x=−3

x−2=0⇒x=2

Final Answer

x=−3,−1,2

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