# Mathematics 1010 online

## Factoring and Solving Polynomial Equations

It's obvious that a polynomial such as

is zero when or . The same fact is less obvious when we rewrite as

In the first form, is said to be factored It is easier to tell the roots of a polynomial if it is factored, but it is also possible to go the other direction: to factor a polynomial when we know one or more of its roots. Just how to go about this is described on this page.

### Knowing a Real Root

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

is zero when . This can be easily checked:

The fact that is a root implies that can be written as where is a polynomial of degree . As discussed elsewhere, can be found by synthetic division :

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

which has the solutions

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 .

### Knowing a Complex Root

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

Then

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

The degree of is less than the degree of .

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

where is a linear polynomial. It can be obtained by long division, and we know from the above discussion that

### A Comprehensive Example

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

and suppose that we wish to find all roots of .

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

is a factor of . We find the quotient by long division :

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

which has the solutions . The roots of our original polynomial are, therefore: