How many times a particular number is a zero for a given polynomial. For example, in the polynomial function f(x)=(x−3)4(x−5)(x−8)2, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2. Although this polynomial has only three zeros, we say that it has seven zeros counting multiplicity.

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Hiroko Cabezas · Accepted Answer

In your example, 3 is a zero of f, because f(3)=0. That&#039;s all it means to call something a zero of a polynomial. If n is some number, and g(x) is a polynomial, we say n is a zero of g(n)=0Now, if we have two polynomials with different zeros, we know the polynomials are different, but how can we tell iff(x)=(x−2)(x−3)andg(x)=(x−2)(x−3)3are different? They both have 2 and 3 as their roots, so that won&#039;t be quite enough to distinguish them. That&#039;s where the multiplicity of a root comes in. We say 3 is a zero of multiplicity 1 for f, whereas 3 is a zero of multiplicity 2 for gAnother way to look at this, is if we look at f(x) without the factor of x-3, namely, the polynomialf′(x)=f(x)x−3=x−2,we can see f′(3)=3−2=1≠0. However, if we look atg′(x)=g(x)x−3=(x−2)(x−3)we have g′(3)=0. So in some sense, the multiplicity can be thought of as &quot;how many times can we remove the zero from the polynomial, until it&#039;s no longer a zero?&quot;Caution: When I write the divisions above, do not try to plug the zeros in to them. I.e., you cannot sayf′(3)=f(3)3−3=00,which I claimed was 1 earlier, as you cannot divide by 0.

Marlene Broomfield · Accepted Answer

In your example, 3 is a zero of f, because f(3)=0. That&#039;s all it means to call something a zero of a polynomial. If n is some number, and g(x) is a polynomial, we say n is a zero of g if g(n)=0Now, if we have two polynomials with different zeros, we know the polynomials are different, but how can we tell iff(x)=(x−2)(x−3)andg(x)=(x−2)(x−3)2are different? They both have 2 and 3 as their roots, so that won&#039;t be quite enough to distinguish them. That&#039;s where the multiplicity of a root comes in. We say 3 is a zero of multiplicity 1 for f, whereas 3 is a zero of multiplicity 2 for g.Another way to look at this, is if we look at f(x) without the factorof x-3 nearly, the polynomialf′(x)=f(x)x−3=x−2we can see f′(3)=3−2=1≠0. However, if we look atg′(x)=g(x)x−3=(x−2)(x−3)we have g′(3)=0. So in some sense, the multiplicity can be thought of as &quot;how many times can we remove the zero from the polynomial, until it&#039;s no longer a zero?&quot;Caution: When I write the divisions above, do not try to plug the zeros in to them. I.e., you cannot sayf′(3)=f(3)3−3=00,which I claimed was 1 earlier, as you cannot divide by 0.

How many times a particular number is a zero for a given polynomial. F

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