SOLUTION: Check that (2+i) is a root of z^4+2(z^3)-9(z^2)-10z+50=0. Find the remaining three roots.

Question 1160754: Check that (2+i) is a root of z^4+2(z^3)-9(z^2)-10z+50=0. Find the remaining three roots.
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

factor
group

use quadratic function to find roots
for

roots are:
->is a root of
since complex roots always come in pairs, you also have

and remaining roots are:

for

roots are:

Answer by MathTherapy(10555) (Show Source): You can put this solution on YOUR website!
Check that (2+i) is a root of z^4+2(z^3)-9(z^2)-10z+50=0. Find the remaining three roots.

As 1 root is 2 + i, its conjugate/other root is: 2 - i
With 2 + i and 2 - i being roots, we get: z = 2 + i and z = 2 - i, and so, FACTORS are: z - 2 - i and z - 2 + i.
The above expands to
Using synthetic division or long division of polynomials, we find that the quotient is: .
Now, since this quotient CANNOT be factored, you can use the quadratic equation formula, or "complete the square" to find the other 2 roots.
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