It's obvious that a polynomial such as

Suppose we recognize, for example by trial and error, or drawing a graph, that the polynomial

This means that

To find the remaining zeros of we only have to find the zeros of . This means we have to solve the quadratic equation

Thus we see that the roots of are and , and *
we were able to find all of them once we recognized one of them*. In
general, if we have a polynomial and one real root we can use
synthetic division to obtain a polynomial of degree one less whose
roots equal the remaining roots of .

Suppose the complex number is a root of a polynomial . Then
we can write
just as above. However, the
coefficients of are complex numbers * even if* the
coefficients of are real numbers. The process still works, but
it requires complex arithmetic, and there is a better way that uses
real arithmetic only.

The key fact in this context is that conjugate
complex of is * also* a root of .
It's a simple exercise to see that this is true by observing that the
conjugate complex of a real number is the number itself, and that for
any complex numbers and

Thus if is a factor of then is also a
factor, and so is the product
. This last term is
a quadratic polynomial whose coefficients are * real*. To see this
let

Hence if we know that is a root of then is also a root and we can obtain a polynomial satisfying

Let's look again at the above example. Essentially we work it backwards. Again, let

We know that is a root of . Thus and So we can write

The following example shows all the principles described above in action. It does not address the question of how one might find a particular root. The answer to that question depends on the context in which the polynomial in question occurs. There are general purpose methods for finding a single root but they are beyond the scope of this class.

Let

It is easily checked that is a root of . This means that is a factor of . We find it by synthetic division :

Hence

Note that the remainder of the division by is zero (underlined in the above table) which confirms that is in fact a root. can be rewritten (after multiplying first factor and dividing the second factor by 2) as

We now need to find the roots of

It turns out that is a root of .

Thus

Again, the fact that the remainder is zero indicates that is actually a factor of . In fact,

The roots of the first factor are , that's how we constructed that factor. The roots of the second factor can be found by solving the quadratic equation