Factor f(x)=3x^3+2x^2-19x+6

Problem

3*x^3+2*x^2−19*x+6

Solution

1. Identify possible rational roots using the Rational Root Theorem by checking factors of the constant term 6 divided by factors of the leading coefficient 3

2. Test the value x=2 using synthetic division or direct substitution.

3. Evaluate ƒ(2)=3*(2)^3+2*(2)^2−19*(2)+6=24+8−38+6=0

4. Divide the polynomial by the factor (x−2) using synthetic division to find the remaining quadratic factor.

5. Result of the division is the quadratic 3*x^2+8*x−3

6. Factor the quadratic 3*x^2+8*x−3 by looking for two numbers that multiply to −9 and add to 8 which are 9 and −1

7. Rewrite the quadratic as 3*x^2+9*x−x−3 and factor by grouping to get (3*x−1)*(x+3)

8. Combine all factors to express the fully factored form of the polynomial.

Final Answer

3*x^3+2*x^2−19*x+6=(x−2)*(3*x−1)*(x+3)

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