Factor x^3+2x^2-5x-6

Problem

x3+2*x2−5*x−6

Solution

1. Identify potential rational roots using the Rational Root Theorem. The possible integer roots are factors of −6 which are ±1,±2,±3,±6

2. Test the value x=−1 by substituting it into the polynomial: (−1)3+2*(−1)2−5*(−1)−6=−1+2+5−6=0 Since the result is zero, (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. Calculate the quotient: (x3+2*x2−5*x−6)÷(x+1)=x2+x−6

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

6. Write the quadratic factor as (x+3)*(x−2)

7. Combine all factors to express the original polynomial in its completely factored form.

Final Answer

x3+2*x2−5*x−6=(x+1)*(x+3)*(x−2)

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