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Unformatted text preview: (a) In this case we’ve got two simple zeroes : x = 5, x = 3 .
(b) Here x = 7 is a zero of multiplicity 2.
(c) There are two zeroes for this polynomial : x = 1 with multiplicity 2 and x = 2 with
multiplicity 3.
(d) We have three zeroes in this case. : x = 5 which is simple, x = 0 with multiplicity of 4 and
x = 3 with multiplicity 3.
(e) In the final case we’ve got four zeroes. x = 5 which is simple, x = 1 with multiplicity of
3, x = 1 with multiplicity 2 and x = 4 which is simple.
This example leads us to several nice facts about polynomials. Here is the first and probably the
most important.
Fundamental Theorem of Algebra
If P ( x ) is a polynomial of degree n then P ( x ) will have exactly n zeroes, some of which may
repeat.
This fact says that if you list out all the zeroes and listing each one k times where k is its
multiplicity you will have exactly n numbers in the list. Another way to say this fact is that the
multiplicity of all the zeroes must add to the degree of the polynomial.
We can go back to the previous example and...
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 Spring '12
 MrVinh

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