SOLUTION: Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree: 18; Zeros: 3 and 1+i. I know that the other zero has to be 1-i, however, I seem to

Question 150527: Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree: 18; Zeros: 3 and 1+i. I know that the other zero has to be 1-i, however, I seem to have misplaced my notes from class and my textbook is not offering much help. Any assistance would be greatly appreciated. Thanks.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Are you sure that the polynomial is not of degree 3? If it's degree 18, then what is the multiplicity of each root?

So I'm going to assume that the polynomial is of degree 3.

Since , , and are given zeros this means that:

, , and

Get all terms to the left side in each case

, , and

Now use the zero product property in reverse to join the factors.

Regroup the terms

Factor to get by use of the difference of squares. Note: let and rewrite the problem into

Rewrite as

Rewrite as

Expand. Remember,

Distribute.

Multiply.

Now combine like terms.

So the polynomial of degree 3 that has the roots , , and is

Notice how if we graph , we can visually verify our answer (note: in this case, we can only verify the root )

Graph of with root of

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