Factor x^3-7x+6

Problem

x3−7*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: 1−7*(1)+6=1−7+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 x3−7*x+6 by (x−1) using synthetic division or long division to find the remaining quadratic factor.

5. Perform the division:

(x3−7*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 numbers are 3 and −2 giving the factors (x+3) and (x−2)

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

Final Answer

x3−7*x+6=(x−1)*(x−2)*(x+3)

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