Find the Roots (Zeros) f(x)=x^3-6x^2+61x-106

Problem

ƒ(x)=x3−6*x2+61*x−106

Solution

1. Apply the Rational Root Theorem to identify potential integer roots from the factors of the constant term −106 The factors are ±1,±2,±53,±106

2. Test potential roots by substituting values into the function. Testing x=2

ƒ(2)=(2)3−6*(2)2+61*(2)−106

ƒ(2)=8−24+122−106

ƒ(2)=0

Since ƒ(2)=0 (x−2) is a factor.

3. Perform synthetic division to divide x3−6*x2+61*x−106 by (x−2)

2∣1−6*61−106

↓2−8*106

1−4*53*0

The resulting quotient is x2−4*x+53

4. Solve the quadratic equation x2−4*x+53=0 using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(−(−4)±√(,(−4)2−4*(1)*(53)))/(2*(1))

x=(4±√(,16−212))/2

x=(4±√(,−196))/2

5. Simplify the complex roots by extracting the imaginary unit i

x=(4±14*i)/2

x=2±7*i

Final Answer

x=2,2+7*i,2−7*i

Want more problems? Check here!