# How do you have multiple multiplicities in a polynomial?

I need to make a 3 degree polynomial with the only zeros being 3 with the multiplicity being "1, 4 and multiplicity of 2." I know that the zeros and 3 degree are typed together like so, (x-3)^3. But how can one have multiple multiplicities?

Here is the full question for future users. "Find a formula for a degree three polynomial function whose zeros are 3 with multiplicity 1, and 4 with multiplicity 2" It was worded funny.

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@Tyler The correct parsing of the sentence is "... whose zeroes are [3 with multiplicity 1], and [4 with multiplicity 2]". So you have two zeroes 3 and 4, and their multiplicities are 1 and 2 respectively. Interpreted this way, the wording is quite normal. –  Ted Oct 30 '12 at 6:17
I agree @Ted, the problem is the brackets are not there for me and I'm not the greatest math whiz, yet ;) –  Tyler Zika Oct 30 '12 at 6:20

The sum of the multiplicities of the roots of a polynomial is equal to the degree of the polynomial (this is essentially the Fundamental Theorem of Algebra). Since your polynomial has degree 3, the only reasonable interpretation of the problem that I can find is that the polynomial should have as roots 3 (with multiplicity 1) and 4 (with multiplicity 2), which identifies the polynomial $c(x-3)(x-4)^2$, where $c$ is any nonzero constant.
To answer your question more directly, there is no such thing as "multiple multiplicities" in the sense of a single root having more than one multiplicity associated with it. The multiplicity of a root $r$ refers to the exponent of $(x-r)$ in the factorization of the polynomial.