Find all of the zeros, real and complex in the equation f(x)= 2x^3 - 4x^2 + 3x - 1

Answer 1

Since f(1) = 0, we know that x=1 is a root. Also, we know that f(x) can be written as (x-1)(ax^2 + bx + c)

Use polynomial division or Synthetic Division to to get the quadratic factor.

(2x^3 - 4x^2 + 3x - 1) / (x - 1) = 2x^2 - 2x + 1

f(x) = ( x - 1 ) ( 2x^2 - 2x + 1 )

To get to the roots of 2x^2 - 2x + 1 = 0, we use the quadratic formula:

x = [ -(-2) +/- sqrt( 2^2 - 4(2)(1) ] / (2*2)

= [ 2 +/- sqrt(-4) ] / 4

= 1/2 +/- i/2

Solution: 1, 1/2 + i/2, 1/2 - i/2