Expert: Paul Klarreich - 4/20/2006

Question
My name is Colleen.  I am studying Pre-calculus.  I am baffled by this problem.  I have been trying to find a similar problem to use as as reference, but have not been successful.  The problem is:
3 and i are two zeros of a cubis function.  What is the third zero?  Find a cubic funstion that has these three zeros.

I appreciate any guidance you can give!
Colleen

Answer
Hi, Colleen,

You wrote:
Subject:  Pre-Calculus
Question:  My name is Colleen. I am studying Pre-calculus. I am baffled by this problem. I have been trying to find a similar problem to use as as reference, but have not been successful. The problem is:
3 and i are two zeros of a cubis function. What is the third zero? Find a cubic funstion that has these three zeros.

I appreciate any guidance you can give!
Colleen
-----------------
You FORGOT to write:

Find a cubic function WITH REAL COEFFICIENTS ...

That is critical.  If the function has real coefficients, then imaginary roots, like x=i, must appear in conjugate pairs.  Therefore,  x = -i must also be a root.  So the three roots are:

x = 3, x = i, x = -i.

And the three factors must be:

(x - 3)(x - i)(x + i)

Now the product of the second and third is  x^2 + 1, so your polynomial is:

(x - 3)(x^2 + 1)

which you can easily multiply out.