SOLUTION: Given 3+i is a root, determine all other roots of f(x)=x^4-2x^3-x^2-38x+130?

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Question 731699: Given 3+i is a root, determine all other roots of f(x)=x^4-2x^3-x^2-38x+130?

Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Another root is 3-i, because complex roots of polynomial functions occur as conjugate pairs. The corresponding quadratic factor for the function based on these two roots would be %28x-%283-i%29%29%28x-%283%2Bi%29%29=x%5E2-6x%2B10 [two steps omitted here].

Now knowing that quadratic factor, divide x%5E4-2x%5E3-x%5E2-38x%2B130 by x%5E2-6x%2B10, polynomial division. That quotient you could then find zeros for using general solution to quadratic equation.

Doing polynomial division for %28x%5E4-2x%5E3-x%5E2-38x%2B130%29+%2F+%28x%5E2-6x%2B10%29=x%5E2%2B4x%2B13, which is not factorable, so....
through general solution to quadratic formula, roots of this are -2-3i and -2+3i.

Summary of the roots:
3+i
3-i
-2-3i
-2+3i