91
C H A P T E R 3
e Arithmetic, Geometry, and
Calculus of Polynomials
Polynomials are a rich family of functions. ey include lines and quadratic equations –
functions that we’ve studied in a good deal of detail because they are useful for so many things.
Polynomials are defined everywhere on the real line; they are continuous, differentiable, and
just generally nice to work with. As we will see in our later study of sequences and series,
polynomials can be used to approximate any continuous function. In this chapter we will learn
to work with polynomials more closely as well as introducing a couple of techniques that are
generally useful in the context of polynomials.
In Section 3.1 we present Newton’s method which is a general technique for finding the roots of
an equation. In Section 3.3 we use polynomials to demonstrate a rule, La’Hospital’s rule, that is
useful for evaluating limits in general. is section should be covered even if the other material
on polynomials is not of interest.
3.1 POLYNOMIAL ARITHMETIC
Consider the problem of computing .x
2
C x C2/
2
. We might do this in the following fashion:
.x
2
C x C2/.x
2
C x C2/ D x
2
.x
2
C x C2/ C x.x
2
C x C2/ C 2.x
2
C x C2/
D x
4
C x
3
C 2x
2
C x
3
C x
2
C 2x C2x
2
C 2x C4
D x
4
C 2x
3
C 5x
2
C 4x C 4
Which is a lot of algebra! If we made the polynomial higher degree it would get worse. Let’s
try this instead.
1 1 2 (1x
2
C 1x C2)
1 1 2 (1x
2
C 1x C2)
2 2 4
1 1 2 .
1 1 2 . .
1 2 5 4 4 (1x
4
C 2x
3
C 5x
2
C 4x4)
92 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
Which also shows .x
2
C x C2/
2
D x
4
C 2x
3
C 5x
2
C 4x C 4. But how? In the “long multi-
plication” table the positions correspond to powers of x. e rightmost place is the 1’s place,
the next is the x’s place, then the x
2
’s place and so on.
It is very similar to the algorithm for multiplying numbers with pencil-and-paper, but there is
no carry. Why no carry? Because when you add up a whole lot of terms of the form ax
m
you
just get many x
m
’s; it never piles up to the point where you get some x
mC1
’s. In other words,
this is actually easier than the multiplication algorithm for numbers.
Knowledge Box 3.1
Fast Polynomial Multiplication
• Place the coefficients of the polynomials in the first two rows.
• In the following rows, scale the coefficients of the first polynomial by each of
the coefficients of the second, one per row.
• e first new row lines up with the ones above it; after that shift left one
place for each row.
• Add up the columns for the rows you just generated.
• is creates a last row that is the coefficients of the product.
Example 3.1 Find
.x
3
C 4x
2
C 2x 3/ .x
2
C 3x C2/
Solution:
1 4 2 -3
1 3 2
2 8 4 -6
3 12 6 -9 .
1 4 2 -3 . .
1 7 16 11 -5 -6
So:
.x
3
C 4x
2
C 2x 3/ .x
2
C 3x C2/ D x
5
C 7x
4
C 16x
3
C 11x
2
5x 6
˙
3.1. POLYNOMIAL ARITHMETIC 93
is technique is enormously faster than imposing the distributive law and collecting terms, and
because it’s structured, it is easier to avoid errors. ere is one potential sand trap – when a term
is missing, you must fill in a zero for it. Let’s do another example that demonstrates this.
Example 3.2 Find .x
2
C x C1/ .x
2
C 4/.
Solution:
1 1 1
1 0 4
4 4 4
0 0 0 .
1 1 1 . .
1 1 5 4 4
So:
.x
2
C x C1/ .x
2
C 4/ D x
4
C x
3
C 5x
2
C 4x C 4
˙
A similar technique – that is more likely to be familiar to you – can also be used to divide
polynomials. It is sometimes called synthetic division.
Example 3.3 Compute .x
3
C 6x
2
C 11x C6/ .x C2/.
Solution:
1 4 3
1 2 1 6 11 6
1 2
4 11 6
4 8
3 6
3 6
0
So:
.x
3
C 6x
2
C 11x C6/ .x C2/ D x
2
C 4x C 3
˙
94 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
ese techniques for multiplication and division of polynomials are essentially book-keeping
devices. ey don’t give you new capabilities, but they make existing capabilities more reliable.
Remember that we often want to find the roots of a function. For instance, when optimizing or
sketching curves, we want to to be able to solve f .x/ D 0, f
0
.x/ D 0, and f
00
.x/ D 0. Being able
to rapidly multiply and divide polynomials makes these tasks easier. e root-factor theorem
tells us that “If f .x/ is a polynomial and f .c/ D 0 for some number c, then .x c/ is a factor
of f .x/.” One thing these new techniques do not do is tell us what value of c to try. Plugging in
values of c and looking for zeros often works – if there is some whole number c that is a root.
Sir Isaac Newton worked out a formula that, given a value close to a root, can move it closer.
Knowledge Box 3.2
Newton’s method for finding roots
If x
0
is close to a root of f .x/, then the sequence generated by using the formula
x
iC1
D x
i
f .x
i
/
f
0
.x
i
/
will approach the nearby root.
Example 3.4 Use Newton’s method with an initial guess of x
0
D 1 to approximate a root of
f .x/ D x
2
2.
Solution:
To start with, we need to find the Newton’s method formula. Since f
0
.x/ D 2x we get
that
x
iC1
D x
i
x
2
i
2
2x
i
Now compute
x
1
D 1
1 2
2
D 1:5
x
2
D 1:5
2:25 2
3
Š 1:4166666667
3.1. POLYNOMIAL ARITHMETIC 95
x
3
Š 1:4142157
x
4
Š 1:4142126
x
5
Š 1:4142126
So by the fifth updating the approximation has stabilized at a value that agrees with
p
2 up
to seven decimals. Since the roots of f .x/ D x
2
2 are ˙
p
2 this is a nice demonstration
that the technique works. Although we are demonstrating Newton’s method on polynomials,
it will work for any function that is continuous and differentiable near a root we are trying to find.
˙
Newton’s method – or other more sophisticated root finding methods – are often built into a
calculator. By showing the values for x
1
, x
2
, and so on, coding the formula into a spreadsheet is
a relatively low work method of doing the calculations. Once you code the formula you can just
let the spreadsheet perform the iterations.
Example 3.5 Find the Newton’s method formula for approximating roots of:
f .x/ D x
3
C 3x 1
Solution:
Just plug into the form for the Newton’s method formula:
x
iC1
D x
i
x
3
i
C 3x
i
1
3x
2
i
C 3
If we start with x
0
D 1:0 we get:
x
0
D1
x
1
D0.5
x
2
D0.36111111
x
3
D0.32917086
x
4
D0.32335786
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