2.5 Fundamental Theorem of Algebra

This theorem is easy to understand: the highest degree in a polynomial function equates to the total number of zeroes that that function has. Before reading this, most likely you were told to find all the real or rational zeroes of a function. Sometimes the only zeroes a function has are real, sometimes not. Sometimes the zeroes are imaginary.

Here is an example:

The given zeroes of x for f(x) are:

A casual glance at this might make one think that this is, at the very least, a cubic function (a polynomial whose highest degree is 3). But that would be a wrong assumption to make. The presence of the 3-i automatically implies the presence of its conjugate, 3+i. So this function is at least a quartic function, and the zeroes actually are:

The easiest function that could be formed from these zeroes would be:

The easiest thing to do would be to then to multiply the lone x and the (x-2) together, and then to multiply the two imaginary zeroes together:

To deal with the imaginary zeroes, we have two options. We could simply distribute all of the variables in the first imaginary zero to the other, but that is tedious and has a high potential for error. Instead, we will ignore the presence of the x's in the imaginary zeroes:

Then, we merely add up these numbers to get a middle term, and then multiply/foil them to get the last term. To demonstrate:

So now our equation looks like this:

All done! One more thing: a problem asking for all of the x-intercepts and a problem asking for all of the zeroes are not asking for quite the same thing, although they're very, very close. Imaginary zeroes do not appear on a normal x-y graph, so when a problem asks for all of the x-intercepts, it is another way of asking for all of the real zeroes. In other words: you don't need to bother finding the imaginary zeroes. But if the problem wants all of the zeroes, then you have to find all of them, both real and imaginary.