Find the Roots (Zeros) f(x)=x^3-3x^2+4x-2

Problem

ƒ(x)=x3−3*x2+4*x−2

Solution

1. Identify a potential rational root using the Rational Root Theorem by testing divisors of the constant term −2

2. Test the value x=1 by substituting it into the function: ƒ(1)=(1)3−3*(1)2+4*(1)−2=1−3+4−2=0

3. Divide the polynomial by the factor (x−1) using synthetic division or long division to find the remaining quadratic factor.

4. Perform the division: (x3−3*x2+4*x−2)÷(x−1)=x2−2*x+2

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

6. Substitute the coefficients a=1 b=−2 and c=2 into the formula: x=(2±√(,(−2)2−4*(1)*(2)))/(2*(1))

7. Simplify the expression: x=(2±√(,4−8))/2=(2±√(,−4))/2=(2±2*i)/2=1±i

Final Answer

x=1,1+i,1−i

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