State the complex and real zeros of this function: X^3-3x^2+x+1.
By simple observation, one root is 1.
Hence (x - 1) is a factor.
Dividng the cubic by this factor gives us,
(x - 1)(x^2 - 2x - 1)
The quadratic x^2 - 2x - 1 becomes (x - 1)^2 - 2 (completing the square)
seting this factor to zero gives us,
(x - 1)^2 - 2 = 0
x = 1 +/- sqrt(2)
The roots are all real: x = 1, x = 1 + sqrt(2), x = 1 - sqrt(2)