Polynomials with Too Few Roots
Okay, we saw that roots of polynomials exactly correspond to linear factors, and that a polynomial can have at most as many roots as its degree. In fact, there’s an expectation that a polynomial of degree will have exactly roots. Today I want to talk about two ways this can fail.
First off, let’s work over the real numbers and consider the polynomial . A root of will be a real number so that , but a little playing around will show us that is the only possible solution. The degree of the polynomial is two, so we expect two roots, but we only find one. What went wrong?
Well, we know from the fact that is a root that is a factor of . Our division algorithm shows us that we can write . The factor that gives us the root shows up twice! But since we’ve already counted the root once the second occurrence doesn’t do anything.
To remedy this, let’s define the “multiplicity” of a root to be the number of times we can evenly divide out a factor of . So in our example the root has multiplicity . When we count the roots along with their multiplicities, we get back exactly the degree.
So do multiplicities handle all our problems? Unfortunately, no. An example here over the real numbers is the polynomial . A root of this polynomial would be a real number with . But since the square of any real number is nonnegative, this can’t be satisfied. So there exist polynomials with fewer roots than their degrees would indicate; even with no roots at all!
Now, some fields are well-behaved. We say that a field is “algebraically closed” if every polynomial over that field has a root . In that case we can divide out by to get a polynomial of one degree less, which must again have a root by algebraic closure. We can keep going until we write the polynomial as the product of a bunch of linear factors — the number is the same as the degree of the polynomial — and leave one field element left once we get down to degree zero. Thus over an algebraically closed field every polynomial has exactly as many roots as its degree indicates.. if you count them with their multiplicities!