The number of roots of a real polynomial function : why is " counting multiplicities" important ? ( Richard Delaware's College Algebra Lessons)

Question

Source : https://www.youtube.com/watch?v=tAfLEocGgDs&list=PLDE28CF08BD313B2A&index=16

• In his college algebra video lessons ( #26) , Pr. R. Delaware states that :

a real polynomial function of degree $n$ has $n$ real or complex roots counting multiplicities

• I think I understand what is the multiplicity of a root : when a root appears in a linear factor ( once the polynomial has been factored) the multiplicity of the root is the exponent of the factor.

• What I don't understand clearly is in which way not counting multiplicities produces ( or can produce) a false result as to the number of roots.

• Could you please provide an example showing the importance of counting multiplicities?

Answer 1

A polynomial function may always have repeated roots. As in $f(x)=(x-1)^2(x-2)^3$. Here the degree is $5$, but, if you don't count multiplicities, there are only $2$ roots.

The fundamental theorem of algebra says that a complex polynomial of degree $n$ has $n$ roots. But this is only true if you count multiplicities.

Answer 2

If you don't count multiplicities, the $x^2$ only has one root. Therefore, if we don't count multiplicities, it will not be true that the degree of the polynomial is equal to the number of roots.