lesson06.pdf

(205 KB) Pobierz
THE UNIVERSITY OF AKRON
Mathematics and Computer Science
mptii
menu
Lesson 6: Dividing & Factoring Polynomials
Directory
• Table of Contents
• Begin Lesson 6
r
IamD S
N ⊆ Z ⊆ Q ⊆ R ⊆ C
a 3 a 4 = a 7 ( ab ) 10 = a 10 b 10
− ( ab − (3 ab − 4))=2 ab − 4
( ab ) 3 ( a − 1 + b − 1 )=( ab ) 2 ( a + b )
( a − b ) 3 = a 3 − 3 a 2 b +3 ab 2 − b 3
2 x 2 − 3 x − 2=(2 x + 1)( x − 2)
1
2 x +13=0 = ⇒ x = − 26
G= { ( x, y ) | y = f ( x ) }
f ( x )= mx + b
y = sin x
Copyright c 1995–2000 D. P. Story
Last Revision Date: 2/2/2000
 
Lesson 6: Dividing & Factoring Polynomials
Table of Contents
6. Dividing & Factoring Polynomials
6.1. Polynomials: A Quick Review
6.2. Polynomial Division
• Polynomial Division Explained
6.3. Factoring Polynomials: Motivation
6.4. Factoring Polynomials: Theory
• Reducible versus Irreducible Polynomials • The Funda-
mental Theorem of Algebra • Roots and LinearFactors Re-
lated
6.5. Factoring Polynomials: Methods
• Factoring x 2 + bx + c • Factoring x 2 − a 2
• Factoring
ax 2 + bx + c • Factoring x 3 ± a 3
6. Dividing & Factoring Polynomials
6.1. Polynomials: A Quick Review
A polynomial in x is an algebraic expression that can be built up
through any (finite) combination of additions, subtractions, and mul-
tiplications of the symbol x with itself and with numerical constants.
The degree of the polynomial is the value of the highest exponent.
Illustration 1. Examples of Polynomials in x .
(a) 2 x +1 has degree 1. A degree one polynomial is sometimes called
linear because its graph is a straight line.
(b) 5 x 2 − 4 x + 3 has degree 2. A degree two polynomial is called a
quadratic polynomial.
(c) 7 x 3 − 4 x 2 − 2
3 x + 8
has degree 3. This is called a cubic poly-
9
nomial.
(d) x 4 + x has degree 4.
(e) x 45 − 5 x 34 + x 3 + 1 has degree 45.
(f) − 3 y 3 +2 y − y + 2 is a polynomial in y of degree 3.
986093714.001.png
Section 6: Dividing & Factoring Polynomials
(g) As a convenience, a constant is considered to be a polynomial
of degree 0. Thus, the algebraic expression 3 may be interpreted
as a polynomial of degree 0.
Sometimes we symbolically denote a polynomial in x by notations
such as P ( x )or Q ( x ); polynomials in some othervariable such as y
would be denoted similarly: P ( y ) and Q ( y ). If P ( x ) is a polynomial
in x , then we write (type) deg( P ( x )) to refer to the degree of P ( x ).
Polynomials arise naturally in many branches of mathematics and
engineering. Polynomials are the basic building blocks used to create
numerical approximations such as the ones used by your hand-held
calculator. The graphs of certain polynomials have properties that
man exploits to create many useful everyday conveniences such as
flashlights and satellite dishes.
One of the reasons for the importance of polynomials—and quotients
of polynomials—is that theirvalues can be computed by elementary
986093714.002.png
Section 6: Dividing & Factoring Polynomials
arithmetic operations: addition, subtraction, multiplication and di-
vision. These are the operations a computer is designed to perform
quickly and e @ ciently.
The sum and product of polynomials is again a polynomial. Illustrate
this assertion by doing the next exercise. Classify each “answer” as a
polynomial and state its degree.
Exercise 6.1. The Algebra of Polynomials. Perform the indi-
cated operations and classify the results.
(a) (4 x 3 − 6 x 2 +2 x +1)+(2 x 3 − 3 x +4)
(b) (7 x 5 − 4 x 3 +12 x − 4) − (5 x 5 +3 x 4 +4 x 3 +2 x − 4)
(c) (3 x − 2)( x 2 − 2 x +1)
(d) ( x 2 − 4)( x 2 +4)
Adding and multiplying polynomials were covered by the general
methods described in Lesson 3 (addition) and in Lesson 5 (mul-
tiplication). In the next paragraph we look at division of polynomials.

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin