In this section we will start looking at polynomials. Polynomials will show up in pretty much every
section of every chapter in the remainder of this material and so it is
important that you understand them.
We will start off with polynomials
in one variable. Polynomials in one
variable are algebraic expressions that consist of terms in the form 
where n
is a non-negative (i.e. positive or
zero) integer and a is a real number
and is called the coefficient of the
term. The degree of a polynomial in one variable is the largest exponent in
the polynomial.
Note that we will often drop the “in one variable” part and
just say polynomial.
Here are examples of polynomials and their degrees.
So, a polynomial doesn’t have to contain all powers of x as we see in the first example. Also, polynomials can consist of a single
term as we see in the third and fifth example.
We should probably discuss the final example a little
more. This really is a polynomial even
it may not look like one. Remember that
a polynomial is any algebraic expression that consists of terms in the form . Another way to write the last example is
Written in this way makes it clear that the exponent on the x is a zero (this also explains the
degree…) and so we can see that it really is a polynomial in one variable.
Here are some examples of things that aren’t polynomials.
The first one isn’t a polynomial because it has a negative
exponent and all exponents in a polynomial must be positive.
To see why the second one isn’t a polynomial let’s rewrite
it a little.
By converting the root to exponent form we see that there is
a rational root in the algebraic expression.
All the exponents in the algebraic expression must be non-negative integers in order
for the algebraic expression to be a polynomial. As a general rule of thumb if an algebraic
expression has a radical in it then it isn’t a polynomial.
Let’s also rewrite the third one to see why it isn’t a
polynomial.
So, this algebraic expression really has a negative exponent
in it and we know that isn’t allowed.
Another rule of thumb is if there are any variables in the denominator
of a fraction then the algebraic expression isn’t a polynomial.
Note that this doesn’t mean that radicals and fractions
aren’t allowed in polynomials. They just
can’t involve the variables. For
instance, the following is a polynomial
There are lots of radicals and fractions in this algebraic
expression, but the denominators of the fractions are only numbers and the
radicands of each radical are only a numbers.
Each x in the algebraic
expression appears in the numerator and the exponent is a positive (or zero)
integer. Therefore this is a polynomial.
Next, let’s take a quick look at polynomials in two variables.
Polynomials in two variables are algebraic expressions consisting of terms
in the form . The degree of each term in a polynomial in
two variables is the sum of the exponents in each term and the degree of the polynomial is the largest
such sum.
Here are some examples of polynomials in two variables and
their degrees.
In these kinds of polynomials not every term needs to have
both x’s and y’s in them, in fact as we see in the last example they don’t need
to have any terms that contain both x’s
and y’s. Also, the degree of the polynomial may come
from terms involving only one variable.
Note as well that multiple terms may have the same degree.
We can also talk about polynomials in three variables, or
four variables or as many variables as we need.
The vast majority of the polynomials that we’ll see in this course are
polynomials in one variable and so most of the examples in the remainder of
this section will be polynomials in one variable.
Next we need to get some terminology out of the way. A monomial
is a polynomial that consists of exactly one term. A binomial
is a polynomial that consists of exactly two terms. Finally, a trinomial is a polynomial that consists of exactly three
terms. We will use these terms off and
on so you should probably be at least somewhat familiar with them.
Now we need to talk about adding, subtracting and
multiplying polynomials. You’ll note
that we left out division of polynomials.
That will be discussed in a later section
where we will use division of polynomials quite often.
Before actually starting this discussion we need to recall
the distributive law. This will be used
repeatedly in the remainder of this section.
Here is the distributive law.
We will start with adding and subtracting polynomials. This is probably best done with a couple of
examples.
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Example 1 Perform
the indicated operation for each of the following.
(a) Add
to . [Solution]
(b) Subtract
from . [Solution]
Solution
(a) Add to .
The first thing that we should do is actually write down
the operation that we are being asked to do.
In this case the parenthesis are not required since we are
adding the two polynomials. They are
there simply to make clear the operation that we are performing. To add two polynomials all that we do is combine like terms. This means that for each term with the same
exponent we will add or subtract the coefficient of that term.
In this case this is,
[Return to Problems]
(b) Subtract from .
Again, let’s write down the operation we are doing
here. We will also need to be very
careful with the order that we write things down in. Here is the operation
This time the parentheses around the second term are
absolutely required. We are
subtracting the whole polynomial and the parenthesis must be there to make
sure we are in fact subtracting the whole polynomial.
In doing the subtraction the first thing that we’ll do is distribute the minus sign through the
parenthesis. This means that we will
change the sign on every term in the second polynomial. Note that all we are really doing here is
multiplying a “-1” through the second polynomial using the distributive
law. After distributing the minus
through the parenthesis we again combine like terms.
Here is the work for this problem.
Note that sometimes a term will completely drop out after
combing like terms as the x did
here. This will happen on occasion so
don’t get excited about it when it does happen.
[Return to Problems]
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Now let’s move onto multiplying polynomials. Again, it’s best to do these in an example.
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Example 2 Multiply
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
This one is nothing more than a quick application of the distributive
law.
[Return to Problems]
(b)
This one will use the FOIL method for
multiplying these two binomials.
Recall that the FOIL method will only work when
multiplying two binomials. If either
of the polynomials isn’t a binomial then the FOIL method won’t work.
Also note that all we are really doing here is multiplying
every term in the second polynomial by every term in the first polynomial. The FOIL acronym is simply a convenient way
to remember this.
[Return to Problems]
(c)
Again we will just FOIL this one out.
[Return to Problems]
(d)
We can still FOIL binomials that involve more than one
variable so don’t get excited about these kinds of problems when they arise.
[Return to Problems]
(e)
In this case the FOIL method won’t work since the second
polynomial isn’t a binomial. Recall
however that the FOIL acronym was just a way to remember that we multiply
every term in the second polynomial by every term in the first polynomial.
That is all that we need to do here.
[Return to Problems]
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Let’s work another set of examples that will illustrate some
nice formulas for some special products.
We will give the formulas after the example.
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Example 3 Multiply
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
(a)
We can use FOIL on this one so let’s do that.
In this case the middle terms drop out.
[Return to Problems]
(b)
Now recall that . Squaring with polynomials works the same
way. So in this case we have,
[Return to Problems]
(c)
This one is nearly identical to the previous part.
[Return to Problems]
(d)
This part is here to remind us that we need to be careful
with coefficients. When we’ve got a
coefficient we MUST do the exponentiation first and then multiply the
coefficient.
You can only multiply a coefficient through a set of
parenthesis if there is an exponent of “1” on the parenthesis. If there is any other exponent then you
CAN’T multiply the coefficient through the parenthesis.
Just to illustrate the point.
This is clearly not the same as the correct answer so be
careful!
[Return to Problems]
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The parts of this example all use one of the following
special products.
Be careful to not make the following mistakes!
These are very common mistakes that students often make when
they first start learning how to multiply polynomials.