Find the Roots/Zeros Using the Rational Roots Test x^3-3x^2-4x+12

Problem

x3−3*x2−4*x+12

Solution

1. Identify the constant term (a_0)=12 and the leading coefficient (a_n)=1

2. List all possible factors of the constant term p ±1,±2,±3,±4,±6,±12

3. List all possible factors of the leading coefficient q ±1

4. Determine the possible rational roots p/q ±1,±2,±3,±4,±6,±12

5. Test the possible roots using synthetic division or substitution. Testing x=2

2−3*(2)2−4*(2)+12=8−12−8+12=0

6. Factor the polynomial using the known root x=2 to get (x−2)*(x2−x−6)

7. Solve the remaining quadratic equation x2−x−6=0 by factoring:

(x−3)*(x+2)=0

8. Identify all roots from the factors: x=2 x=3 and x=−2

Final Answer

x3−3*x2−4*x+12=0⇒x={2,3,−2}

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