Module 3 
3.3
(see Text sections 3.2, 3.3, and 3.4)
Introduction
We’ve seen that factoring can help us
understand the behaviour and graphs of polynomial functions. One way to factor
numbers is by trying division by potential prime factors, and the same is true
of polynomials. In this section we study
how to divide with polynomials using a pattern very similar to the “long
division” we use for numbers. When the division of numbers does not give an
exact integer result we are led to consider fractions or in other words rational numbers. Similarly,
when the result of dividing two polynomials does not give a polynomial result,
we call the result a rational function.
Such functions will be studied algebraically in this section, and graphically
in the next one.
Although most of the topics in sections 3.2, 3.3 and 3.4 of the text are potentially useful and interesting, some of them are unlikely to be needed in your calculus course and so you will not be required to learn them in this course. So in this section you might find it more efficient to work with these notes and the text at the same time, and read the various pages of the text only as and when they are referred to in the notes.
Long Division of Polynomials
You definitely will need to be able to carry out the “long division” algorithm, so read pages 219-222 carefully.
The statement in the blue box on page 220 can be written
also as
.
So the result of the division of f(x)
by p(x) is to express the ratio as the sum of a polynomial q(x)
(called the “quotient”), and another ratio in which the numerator r(x)
is of lower degree than the denominator p(x).
The Remainder and Factor Theorems
Q1
Note that the Remainder Theorem can be used
in two ways. One, as in Example#1 on page 221, is to use the remainder after
long division of f(x)
by x-a to evaluate f(a).
The other would be to go the other way and
find the remainder without actually doing the division. For example, if
we want to find the remainder on dividing
by x-2,
we need only evaluate
(which is a lot quicker than actually going
through the whole long division process). What is the remainder on dividing
by
? (See answer #1)
In particular, if ,
then the remainder on dividing
by x-a is also 0, so x-a is a factor in
.
This is the Factor Theorem.
A consequence of the Factor Theorem is that
a polynomial of degree n with n
specified roots, say at ,
must be of the form
for some constant a. You may recall that we used the special case of this with n=2 to find the quadratic with specified
intercepts in the first part of this Module
(See Q3 in Module 3.1).
If the quotient on dividing by
also has
as a factor, then
divides into
.
The highest power of
which does so is called the multiplicity of the root at x = r.
At a root of odd multiplicity the graph crosses the x-axis, but at a root of even multiplicity it does not. Either way,
if the multiplicity is greater than one, then the graph is tangent to the x-axis at the point of contact.
Synthetic Division (Optional)
The method of “Synthetic Division”
discussed on pages 222-225 is optional and not recommended. It only works when
the divisor is of the form ,
and because it hides the reasoning behind the process it makes it hard for you
to find and correct errors. It may be a bit quicker, but if anyone really
wanted to do a lot of these calculations they’d use a computer or programmable
calculator.
Complex Roots (Optional)
The material on complex roots in sections 3.3 and 3.4 will not be needed in your calculus course, so you do not need to read it very carefully right now. But you should be aware that a polynomial of degree n cannot have more than n real roots.(It can have less though. The “Theorem on the Exact Number of Zeros” on page 231 is talking about complex roots and for an even degree polynomial it is possible that none of them are real.)
Graphs and (Real)Roots
The Examples#1&2 on pages 229-231 and the pictures on page 230 are worth looking at. In particular, Example 1 on page 229 illustrates our comment above about using the Factor Theorem to determine the formula for a polynomial from the intercepts of its graph, and our comment about multiplicities is illustrated on page 231.
The material on pages 234-237 can help you decide where to look for roots of a polynomial, but in practice a computer generated graph will do the job.
Optional
Section 3.4 is also optional extra reading. The theorem on rational zeros can be used to limit the numbers worth trying if you are looking for an exact root by trial and error, but in practice there is no reason why a polynomial arising in an applied problem should actually have rational roots, so the rational roots theorem doesn’t help much.
Finding Formulas From Graphs
In Example 2 of Section 3.3 on page 231, the equation is used to find the graph, but using the idea of Example 1 we could go the other way and use the graph to find the formula (much as we did for linear and quadratic examples in Section 3.1).
In fact, on the graph in Figure 1 we see
zeros at -1 (where the multiplicity must be even), at 2 (of multiplicity 1) and
at 4 (with odd multiplicity greater than 1). Thus the minimum total
multiplicity is 6, and if the function has degree 6 it must be of the form .
Using f(0)=8, we can solve to get a=1/16 and are done.
Q2
Try using the same idea to determine the polynomial of lowest degree whose graph matches the one below. (See answer #2)
Further Practice
Problem types that you should be able to solve in sections 3.2,3,and 4 are as follows:
Section 3.2 #1-20 and 39-54; Section 3.3 #1-4,7-22,and 45-52.;Section3.4#33-38
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort)
As a minimum try Sec3.2#3,13,19,49;Sec3.3#9,13,19,49;Sec3.4#33,35.
If you need to refer to the study guide on any of these, do so, but then try to do another question of the same type without using the guide.