Polynomials - University of Adelaide

30 Pages·2013·4.45 MB·English

MathsTrack

(NOTE Feb 2013: This is the old version of MathsTrack.

New books will be created during 2013 and 2014)

MToodpuicle 19

Introduction to Matrices
Polynomials

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MATHEMATICS LEARNING SERVICE

CMentrAe foTr LHearSningL anEd AProRfessNionIaNl DeGvelopCmeEntN TRE
Level 1, Schulz Building (G3 on campus map)
Level 3, Hub Central, North Terrace Campus, The University of Adelaide
TEL [phone] | FAX [phone] | [email protected]
TEL [phone] — FAX [phone] — [email protected]
www.adelaide.edu.au/clpd/maths/
www.adelaide.edu.au/mathslearning/
This Topic . . .
Many real situations can be represented by mathematical models that are built from
three kinds of elementary functions. These functions are:
• algebraic functions
• exponential and logarithmic functions
• trigonometric functions
The most common type of algebraic function is the polynomial. This module re-
vises and explores polynomial functions. Later, calculus will be used to investigate
polynomials further.
The Topic has 2 chapters:
Chapter 1 begins by revising the algebra of polynomials. Polynomial division and
the remainder theorem are introduced. The relationship between the zeros
and factors of a polynomial is explored, and the special case of polynomials
with integer coefficients is considered.
Chapter 2 explores the graphs of polynomial functions. It begins by examining
howtheleadingtermofthepolynomialinfluencestheglobalshapeofitsgraph,
and then explores how the factors of the polynomial influence the local shape
of its graph.
Auhor: Dr Paul Andrew Printed: February 24, 2013
i
Contents
1 The Algebra of Polynomials 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Combining polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Division of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Zeros and Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Graphs of Polynomials 14
2.1 Continuity and Smoothness . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Shapes of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Selected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
A Factorisation (revision) 25
A.1 Simple factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A.2 The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B Answers 27
ii
Chapter 1
The Algebra of Polynomials
1.1 Introduction
The most common type of algebraic function is a polynomial function.1 A poly-
nomial in x is a sum of multiples of powers of x. If the highest power of x is n, then
the polynomial is described as having degree n in x.
Definition
A general polynomial of degree n in the variable x is written as
a xn +a xn−1 + ... +a x2 +a x+a
n n−1 2 1 0
where a (cid:54)= 0. The numbers a are called coefficients, with a the leading coeffi-
n i n
cient and a the constant term.2
0
It is common to use subscript notation a for the coefficients of general polynomial
i
functions, but we use the simpler forms below for polynomials of low degree.
Zero degree: f(x) = a Constant function
First degree: f(x) = ax+b Linear function
Second degree: f(x) = ax2 +bx+c Quadratic function
Third degree: f(x) = ax3 +bx2 +cx+d Cubic function
Example
linear
quadratic a. The linear function
cubic 2x+3
quartic
quintic has degree 1, leading coefficient 2 and constant term 3.
.
.
.
1Poly-nomial means ‘many terms’.
2It is called the constant term as it doesn’t change when the variable is given different values.
1
2 CHAPTER 1. THE ALGEBRA OF POLYNOMIALS
b. The quadratic function
x2 −2x
has leading coefficient 1 and no constant term. The coefficient of x is −2.
c. The cubic polynomial
−t3 +3t−17
has leading coefficient −1 and constant term −17. The coefficient of t2
is 0.
d. The polynomials
h4 +3h3 −7h2 +h−10 and −3+4L+2L2 +L5
are examples of a quartic polynomial (degree 4) in h and a quintic
polynomial (degree 5) in L. Polynomials are written with powers either
descending or ascending. The leading coefficient is 1 for each polynomial.
Note: if some of the exercises below seem a bit tricky, then see MathsStart Topic 3:
Quadratic Functions.
Exercise 1.1
1. A simple cost function for a business consists of two parts
• fixed costs which need to be paid no matter how many items of a product
are produced (eg. rent, insurance, business loans, etc).
• variable costs which depend upon the number of items produced.
A computer software company produces a new spreadsheet program which
costs $25 per copy to make, and the company has fixed costs of $10,000 per
month. Find the total monthly cost C as a function of the number x of copies
of the product made.
2. The diagram represents a rectangular garden which is enclosed by 100 m of
fencing.
x
Show that
a. the area A of the garden is is given by the quadratic function
A = 50x−x2
where x m is the length of one of the sides of the garden.
b. the area is a maximum when the garden is square.
1.1. INTRODUCTION 3
3. The graphs of
y = x2 −4x and y = 2x−x2
are sketched below.
y
0 2 4 x
Prove that
a. the graphs meet when x = 0 and x = 3
b. The vertical separation between the graphs is
6x−2x2
for 0 ≤ x ≤ 3.
c. The vertical separation is a maximum when x = 11.
2
4 CHAPTER 1. THE ALGEBRA OF POLYNOMIALS
1.2 Combining polynomials
Many functions in mathematics are constructed out of simpler ‘building-block’ func-
tions. In this section we consider some of the ways polynomials can be combined to
obtain new functions.
If f and g are two functions and c is a fixed number, then we can construct new
functions using the sum f +g, the difference f −g, the (scalar) multiple cf, the
product f ·g and the quotient f/g.
1. Sums and Differences
Polynomials are added (or subtracted) by adding (or subtracting) like terms.3
Example
adding &
subtracting a. If f(x) = 2x2 +x−3 and g(x) = −x2 +4x+5, then
f(x)+g(x) = (2x2 +x−3)+(−x2 +4x+5) = x2 +5x+2
b. If p(t) = 4−5t and q(t) = 2+t−2t2, then
p(t)−q(t) = (4−5t)−(2+t−2t2) = 2−6t+2t2
2. (Scalar) Multiples
When a polynomial is multiplied by a number or a constant, each term is multiplied
by that number.4
Example
multiplying
by a number a. If f(x) = 2x2 +x−3, then 10f(x) = 20x2 +10x−30
or a constant b. If p(t) = 4−5t and q(t) = 2+t−2t2, then
2p(t)−3q(t) = 2(4−5t)−3(2+t−2t2) = 2−13t+6t2
3. Products
When one polynomial is multipled by another polynomial, each term in one poly-
nomial is multiplied by each term in the other.
Example
multiplying
If f(x) = 2x2 +x−3 and g(x) = −x2 +4x+5, then
polynomials
f(x)g(x) = (2x2 +x−3)(−x2 +4x+5)
= 2x2(−x2 +4x+5)+x(−x2 +4x+5)−3(−x2 +4x+5)
= (−2x4 +8x3 +10x2)+(−x3 +4x2 +5x)+(3x2 −12x−15)
= −2x4 +7x3 +17x2 −7x−15
3These are terms that have the same power of the variable.
4The number sometimes called a scalar.
1.2. COMBINING POLYNOMIALS 5
4. Quotients
A function expressed as the quotient of two polynomials is called a rational func-
tion. The domain of a rational function excludes all values of the variable for
which the denominator is zero.
Example
rational
If p(x) = x−3 and q(x) = x+5, then
functions
p(x) x−3
=
q(x) x+5
provided x (cid:54)= −5.
This function can also be represented as
8
1−
x+5
x−3 8 8
In this example, and are rational functions but 1− is
x+5 x+5 x+5
not.
Exercise 1.2
1. If p(x) = 2x − 3 and q(x) = x2 − 4x + 5, simplify the expressions below by
expanding brackets and collecting like terms:
a. 5p(x)+3q(x)
b. xp(x)−2q(x)
c. p(x)q(x)
d. p(x)2
2. A manufacturer plans to make cake tins from 40 cm by 50 cm rectangular
metal sheets. Squares will be cut from each corner and the metal will be
folded as in the diagram below. The volume of the cake tin will depend upon
its height.
50 cm
40 cm
x cm
6 CHAPTER 1. THE ALGEBRA OF POLYNOMIALS
a. Show that
i. If the height of the cake tin is x cm, then the volume is
V = x(40−2x)(50−2x) cm3.
ii. ...and that this is the cubic polynomial
V = 4x3 −180x2 +2000x cm3.
b. What are the restrictions on the x values?
1.3. DIVISION OF POLYNOMIALS 7
1.3 Division of polynomials
When we divide 47 by 3 we get “15 with 2 left over”, and write
47 2
= 15+
3 3
This shows that 47/3 can be written as the sum of a whole number and a simple
fraction between 0 and 1. The number 3 is called the divisor, the number 15 is
called the quotient5 and 2 is called the remainder.
As we shall see soon, when we divide 2x2 +x+1 by x+2 we obtain:
2x2 +x+1 7
= 2x−3+
x+2 x+2
Here x+2 is called the divisor, 2x−3 is the quotient and 7 is the remainder.
This shows that the rational function
2x2 +x+1
x+2
canberepresentedasthesumofthepolynomial2x−3andasimplerationalfunction.
The rational function
7
x+2
is considered to be a simple because the degree of the polynomial in the numerator
is less than the degree of polynomial in the denominator. It can not be reduced to
a simpler function.
This representation is important as it shows that
2x2 +x+1
≈ 2x−3
x+2
when x has large values, since when x is large, then 7 is very small.
x+2
The algorithm6 for dividing one polynomial by another is similar to the algorithm
for dividing whole numbers.
5The word quotient has two meanings (i) the number that results from the division of one
number by another, and (ii) the whole number part of the result of dividing one number by
another. The second meaning is used in this section
6Analgorithmisastep-by-stepprocedureforsolvingamathematicalprobleminafinitenumber
of steps, often involving repetition of the same basic operation. Algorithms are frequently used to
solve mathematical problems on computers.