Find the Roots/Zeros Using the Rational Roots Test x^4-2x^3+x^2-8x-12

Problem

x4−2*x3+x2−8*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 and the leading coefficient q

p:±1,±2,±3,±4,±6,±12

q:±1

3. Determine the possible rational roots p/q by dividing the factors of p by the factors of q

p/q:±1,±2,±3,±4,±6,±12

4. Test the possible roots using synthetic division or substitution. Testing x=−1

(−1)4−2*(−1)3+(−1)2−8*(−1)−12=1+2+1+8−12=0

So, x=−1 is a root.

5. Perform synthetic division with x=−1 to reduce the polynomial.

(x4−2*x3+x2−8*x−12)÷(x+1)=x3−3*x2+4*x−12

6. Test the remaining possible roots on the quotient x3−3*x2+4*x−12 Testing x=3

(3)3−3*(3)2+4*(3)−12=27−27+12−12=0

So, x=3 is a root.

7. Perform synthetic division with x=3 on the reduced polynomial.

(x3−3*x2+4*x−12)÷(x−3)=x2+4

8. Solve the remaining quadratic equation x2+4=0 to find the final roots.

x2=−4

x=±2*i

Final Answer

x=−1,3,2*i,−2*i

Want more problems? Check here!