Roots of a Polynomial

Question

Why is it important to consider multiplicity when determining the roots of a polynomial equation? Example?

Answer 1

It's a great question. A theorem discovered by Gauss says that any polynomial function of degree N has exactly N zeros in the complex number system. This is universally true only when you count up "Multiplicity." For example, consider the 3rd-degree polynomial function (given in factored form)

f(x) = (x–3)²(x+1) .

The degree of this polynomial is 3, since, if you multiply it out, the highest power of x is 3. Put that into a graphing utility, and notice that the graph crosses the x-axis x = –1 and touches at x = 3. So in a way that sort of makes it look like you have two total zeros. But, setting the function equal to zero,

(x–3)²(x+1) = 0 ,

and rewriting in a more expanded form,

(x–3)(x–3)(x+1) = 0 ,

you see that there are exactly three solutions to the equation -- not three distinct solutions, though, because "3" solves the equation f(x) = 0 twice, so "3" is a zero with multiplicity = two . The number of zeros always equals the number of linear factors, even though sometimes you get repetition, or multiplicity.

Answer 2

I hope this helps! Message if you have any more questions.