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

Problem

x3−2*x2−5*x+6

Solution

1. Identify a possible root using the Rational Root Theorem by testing factors of the constant term 6

2. Substitute x=1 into the expression to check if it is a root: 1−2*(1)2−5*(1)+6=1−2−5+6=0

3. Apply the Factor Theorem, which states that since x=1 is a root, (x−1) is a factor of the polynomial.

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

5. Calculate the quotient: (x3−2*x2−5*x+6)÷(x−1)=x2−x−6

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

7. Determine the factors of the quadratic: (x−3)*(x+2)

8. Combine all factors to write the completely factored form of the original cubic expression.

Final Answer

x3−2*x2−5*x+6=(x−1)*(x−3)*(x+2)

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