Factor x^3-4x^2+x+6

Problem

x3−4*x2+x+6

Solution

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

2. Test the value x=−1 by substituting it into the polynomial.

(−1)3−4*(−1)2+(−1)+6=−1−4−1+6=0

Since the result is 0 (x+1) is a factor.

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

(x3−4*x2+x+6)/(x+1)=x2−5*x+6

4. Factor the resulting quadratic expression x2−5*x+6 We look for two numbers that multiply to 6 and add to −5 These numbers are −2 and −3

x2−5*x+6=(x−2)*(x−3)

5. Combine all the factors to write the completely factored form of the cubic polynomial.

(x+1)*(x−2)*(x−3)

Final Answer

x3−4*x2+x+6=(x+1)*(x−2)*(x−3)

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