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