Answer:
x = 3 + i, x = 3 - i , x = 5 , and x = -1
Explanation:
We know that given the polynomial with real coefficients, if x = 3 + i is a root, then x = 3 - i must be another root. These two roots can combine multiplicative factors as shown:
(x - 3 - i) * (x - 3 + i) = (x-3)^2 + 1 = x^2 - 6 x + 9 + 1 = x^2 - 6 x + 10
Then we divide the given polynomial: x^4 - 10x^3 + 29x^2 - 10x - 50 by the quadratic one we just found, and get a perfect division:
{x^4 - 10x^3 + 29x^2 - 10x - 50} / {x^2 - 6 x + 10} = x^2 - 4 x - 5
which can be factored out by grouping as shown below:
x^2 - 4 x - 5 = x^2 - 5 x + x - 5 = x (x - 5) + x - 5 = (x - 5 ) * (x + 1)
then the full factor form of the quartic polynomial given is:
(x - 3 - i) * (x - 3 + i) * (x - 5 ) * (x + 1)
and therefore the associated roots are:
x = 3 + i, x = 3 - i , x = 5 , and x = -1