We’ll start off this section by defining just what a root or zero of a polynomial is. We
say that is a root or zero of a polynomial, ,
if . In other words, is a root or zero of a polynomial if it is a
solution to the equation .
In the next couple of sections we will need to find all the
zeroes for a given polynomial. So,
before we get into that we need to get some ideas out of the way regarding
zeroes of polynomials that will help us in that process.
The process of finding the zeros of really amount to nothing more than solving the
equation and we already know how to do that for second
degree (quadratic) polynomials. So, to
help illustrate some of the ideas were going to be looking at let’s get the
zeroes of a couple of second degree polynomials.
Let’s first find the zeroes for . To do this we simply solve the following
So, this second degree polynomial has two zeroes or roots.
Now, let’s find the zeroes for . That will mean solving,
So, this second degree polynomial has a single zero or
root. Also, recall that when we first
looked at these we called a root like this a double root.
We solved each of these by first factoring the polynomial
and then using the zero factor property on
the factored form. When we first looked
at the zero factor property we saw that it said that if the product of two
terms was zero then one of the terms had to be zero to start off with.
The zero factor property can be extended out to as many
terms as we need. In other words, if
we’ve got a product of n terms that
is equal to zero, then at least one of them had to be zero to start off
with. So, if we could factor higher
degree polynomials we could then solve these as well.
Let’s take a look at a couple of these.
Example 1 Find
the zeroes of each of the following polynomials.
In each of these the factoring has been done for us. Do not worry about factoring anything like
this. You won’t be asked to do any
factoring of this kind anywhere in this material. There are only here to make the point that
the zero factor property works here as well.
We will also use these in a later example.
Okay, in this case we do have a product of 3 terms however
the first is a constant and will not make the polynomial zero. So, from the final two terms it looks like
the polynomial will be zero for and . Therefore, the zeroes of this polynomial
We’ve also got a product of three terms in this
polynomial. However, since the first
is now an x this will introduce a
third zero. The zeroes for this
because each of these will make one of the terms, and
hence the whole polynomial, zero.
With this polynomial we have four terms and the zeroes
Now, we’ve got some terminology to get out of the way. If r
is a zero of a polynomial and the exponent on the term that produced the root
is k then we say that r has multiplicity k. Zeroes with a multiplicity of 1 are often
called simple zeroes.
For example, the polynomial will have one zero, ,
and its multiplicity is 2. In some way
we can think of this zero as occurring twice in the list of all zeroes since we
could write the polynomial as,
Written this way the term shows up twice and each term gives the same
zero, . Saying that the multiplicity of a zero is k is just a shorthand to acknowledge
that the zero will occur k times in
the list of all zeroes.
This example leads us to several nice facts about
polynomials. Here is the first and
probably the most important.
Theorem of Algebra
This fact says that if you list out all the zeroes and
listing each one k times where k is its multiplicity you will have
exactly n numbers in the list. Another way to say this fact is that the
multiplicity of all the zeroes must add to the degree of the polynomial.
We can go back to the previous example and verify that this
fact is true for the polynomials listed there.
This will be a nice fact in a couple of sections when we go
into detail about finding all the zeroes of a polynomial. If we know an upper bound for the number of
zeroes for a polynomial then we will know when we’ve found all of them and so
we can stop looking.
The next fact is also very useful at times.
The Factor Theorem
Again, if we go back to the previous example we can see that
this is verified with the polynomials listed there.
The factor theorem leads to the following fact.
There is one more fact that we need to get out of the way.
This fact is easy enough to verify directly. First, if is a zero of then we know that,
since that is what it means to be a zero. So, if is to be a zero of then all we need to do is show that and that’s actually quite simple. Here it is,
and so is a zero of .
Let’s work an example to see how these last few facts can be
of use to us.
Example 3 Given
that is a zero of find the other two zeroes.
First, notice that we really can say the other two since
we know that this is a third degree polynomial and so by The Fundamental
Theorem of Algebra we will have exactly 3 zeroes, with some repeats possible.
So, since we know that is a zero of the Fact 1 tells us that we can write as,
and will be a quadratic polynomial. Then we can find the zeroes of by any of the methods that we’ve looked at
to this point and by Fact 2 we know that the two zeroes we get from will also by zeroes of . At this point we’ll have 3 zeroes and so we
will be done.
So, let’s find .
To do this all we need to do is a
quick synthetic division as follows.
Before writing down recall that the final number in the third
row is the remainder and that we know that must be equal to this number. So, in this case we have that . If you think about it, we should already
know this to be true. We were given in
the problem statement the fact that is a zero of and that means that we must have .
So, why go on about this?
This is a great check of our synthetic division. Since we know that is a zero of and we get any other number than zero in
that last entry we will know that we’ve done something wrong and we can go
back and find the mistake.
Now, let’s get back to the problem. From the synthetic division we have,
So, this means that,
and we can find the zeroes of this. Here they are,
So, the three zeroes of are ,
As an aside to the previous example notice that we can also
now completely factor the polynomial . Substituting the factored form of into we get,
This is how the polynomials in the first set of examples
were factored by the way. Those require
a little more work than this, but they can be done in the same manner.