SOLUTION: Please help me with this equation-Thank You! Find a polynomial equation with real coefficients that has the given roots. 3, -9, 3+2i

Question 135278: Please help me with this equation-Thank You!
Find a polynomial equation with real coefficients that has the given roots.
3, -9, 3+2i

Found 2 solutions by scott8148, solver91311:
Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
if r is a root, then x-r is a factor

complex roots (3+2i) occur in conjugate pairs, so 3-2i is also a root

(x-3)(x+9)(x-3-2i)(x-3+2i)=0

multiply the factors to find the equation
__ HINT: start with the complex factors, it will be less "messy"
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Two things:
First, if a polynomial equation has a complex number root (), then it also has the conjugate of that complex number as a root (). That means that, although you were only given three numbers as roots, there are actually four, namely:
, , , and

Second, a polynomial equation has a root if and only if is a factor of the polynomial.

So if the desired polynomial is , then . All you need to do now is multiply the factors and collect like terms. Hint: Remember when you are working it out that , so not .

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