Daniel B. answered 09/14/21
A retired computer professional to teach math, physics
FIRST ROOT:
Factor the polynomial in the following steps:
x4 -5x³ + 6x² + 4x - 8
x4 - 2x³ - 3x³ + 6x² + 4x - 8
x³(x - 2) - 3x²(x - 2) + 4(x - 2)
(x³ - 3x² + 4)(x - 2)
This gives the real root of x = 2 and leave us with the polynomial x³ - 3x² + 4 for the other roots.
SECOND ROOT:
Just by looking at the polynomial x³ - 3x² + 4 is it easy to guess the root x = -1.
Polynomial division then leaves us with the polynomial x² + 4x + 4 for the other roots.
THIRD AND FOURTH ROOT:
The quadratic formula gives us x = -2 as a double root of x² + 4x + 4.