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 multiplic