Factoring over the Complex Numbers, Answer 7

Exercise 7.

Give an example of a polynomial of degree 5, whose only real roots are x=2 with multiplicity 2, and x=-1 with multiplicity 1.

Answer.

We need factors of the form

and (x+1) to satisfy the root requirements. We also know that the last 2 of the 5 roots of the polynomial have to be complex; so, for instance,

will do:

The polynomial

satisfies all the requirements.

N.B.: In problems like this one, do not bother to multiply out. You were just asked to write down a polynomial with certain properties; nobody told you to write down the polynomial in a particular form.

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Tue Jun 24 09:53:41 MDT 1997