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

Problem

x3+3*x2−x−3

Solution

1. Identify the constant term (a_0)=−3 and the leading coefficient (a_n)=1

2. List all possible factors of the constant term p∈{±1,±3}

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

4. Determine the possible rational roots p/q by dividing the factors of p by the factors of q which gives the set {±1,±3}

5. Test the possible roots using synthetic division or substitution. For x=1

1+3*(1)2−(1)−3=1+3−1−3=0

6. Factor the polynomial using the known root x=1 to get (x−1)*(x2+4*x+3)

7. Solve the remaining quadratic equation x2+4*x+3=0 by factoring it into (x+1)*(x+3)=0

8. Identify all roots from the factors: x=1 x=−1 and x=−3

Final Answer

x=1,−1,−3

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