E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals B.1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

B.1


Polynomials

KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

Definitions Polynomial Evaluation Polynomial Equations Algebra of polynomials Factoring Polynomials


Definitions:

An expression containing a real number a multiplied by one or more variable raised to some non-negative powers is called a term with coefficient a. For example

Terms which contain the same variable raised to the same powers are called like terms.


are terms, with coefficients and6 2 5 3 1 1 respectively, , , ,

2 2 3 26 2 5 3, , , , and x x x y z wu


Definitions:The degree of a term consisting of a constant

a and no variables is zero if a ≠ 0, and undefined if a =0. The degree of a term which is a product of a constant a and one or more variables is the sum of the powers of the variables. For example

The degree of –2 is zero The degree of 0 is undefined The degree of is 6 The degree of is 4.


4 23x y22 abc


Note:


Two like terms are added using the distribute law, multiplied and divided using the laws of exponents. For example

Example 12 2 21. 4 3 . ax w ax w ax w

4

2 3 52. 28 7 .4

aba b a b

2 42

2

243. 4

6

x yzxz

xyz


Definition

An algebraic expression consisting of a finite sum of terms in the same variable(s) is called a polynomial in those variables. The greatest degree of any term in a polynomial is called the degree of the polynomial. For example,

A polynomial consisting of one, two, or three terms is called a monomial, binomial or trinomial, respectively.



Polynomials in One Variable


11 1 0... ; 0,

n nn n np x a x a x a x a a

Where 1 1 0, ,..., ,n na a a a are the coefficients of

p x . The term n

na x is called the leading term,

and na is called the leading coefficient of

p x .

A polynomial p(x), of degree n in x consist, in general, n+1 terms and can be written



Example 1

In the following table we identify the degree, the coefficients and the leading term for the given polynomials.

2 2

5 3 5

6 0 6 6

3 5 1 3,

( ) De

5 3

5 6

gree Coefficients

8 2 5, 6,8 2

3 9 2 5 1, 3, 9,2

Leading Term

x x

x x x

x x

p x

x x


Polynomial Evaluation


4 3

Consider the polynomial

3 20 3 p x x x x

4 3

4 3

4 3

3 20 3 3

3 20

0 0 0 0

1 1 1 1

2 2 2

3 19

3 20 3 772

p

p

p


Algebra of Polynomials


Addition and subtraction of polynomials are defined using commutative, associative and performed by combining the like terms in the resulting expressions.

Example 2

Let 23 4, 2 5 p x x q x x x .

Find and . p x q x p x q x



Example 2

23 4 2 5p x q x x x x 23 4 2 5x x x

2 3 2 4 5x x x 2 1x x

23 4 2 5p x q x x x x 23 4 2 5x x x

2 3 2 4 5x x x 2 5 9x x


Scalar Multiplication


Scalar multiplication of a polynomial

11 1 0...n n

n np x a x a x a x a

of degree n by a number s ≠ 0 is defined as

11 1 0...n n

n np x as s sx a x a x as s

which results in a polynomial of degree n, and

coefficients 1 1 0, ,..., ,n na a as s s sa



Example 3

Let 3 24 6 5 2 and 2.p x x x x s Find the polynomial sp(x).

Solution

3 22 4 2 6 2 5 2sp x x x

3 28 12 10 4sp x x x x


Multiplication of Polynomials


Multiplication of polynomials p (x) and q (x) of degrees n and m is defined using the distributive, commutative, and associative laws of numbers, and performed by combining the like terms. This leads to a polynomial of degree n + m.

Example 4

2Let 2 5, 5 3 9, find p x x q x x x p x q x



Solution

22 5 5 3 9 p x q x x x x

2 22 5 (2 ) 3 2( 9) 5 (5 ) 5 3 5 9x x x x x x

3 2 210 6 18 25 15 45x x x x x

3 210 19 33 4x x x


Special Products


2 2 2( ) 2i a b a b a b a ab b

2 2 22a b a b a b a ab b

3 2 2( ) 2ii a b a ab b a b 3 2 2 33 3a a b ab b

3 2 22a b a ab b a b

3 2 2 33 3a a b ab b



Special Products

2 2( )iii a b a b a b

2 2 3 3( ) ( )iv a b a ab b a b

2 2 3 3( ) ( )v a b a ab b a b



Example 4

2 21) 2 5 4 20 25t t t

3 2 32) 2 3 8 36 54 27z z z z

3 3 63) 2 2 4x x x

2 34) 5 2 25 10 4 125 8x x x x


Factoring of Polynomials


Factoring polynomials is the reverse operation to multiplication. It consists of writing the given polynomial as a product of polynomials (generally of lower degrees).

When factoring a polynomial, we must first factor out the greatest common factor ( GCF) of the terms of the polynomial. This operation uses the distributive law in reverse manner.



Example 5

Factor out the greatest common factor of the terms of the following polynomial expressions.

2) 25 40a a a 5 (5 8)a a

2 5 5 3) 3 6b x y x y 2 3 2 33 2x y y x

7 44) 2 2 6 2c x y x y 4 332 2 2 3x y x y



Example 6

Factor using the special products (i) –( iv) in reverse manner (after you factor out the GCF, if any)

2. 4 12 9 a x x 22 3 x

4 4. 81 b x y 2 2 2 29 9 x y x y

2 23 3 9 x y x y x y2 2 2

.16 9

z y w

c2 3 2 3

zy w zy w



3 3)125 64d x y 2 25 4 25 20 16x y x xy y

6) 3 192e x 63 64x 3 33 8 8x x

2 23 2 2 4 2 2 4x x x x x x

6 3) 4 4f x x 23 2x 22 22 2 4x x x


Factoring by Grouping


When factoring a polynomial expression of four or more terms, it may happen that grouping the terms into groups and factoring each group such that a common factor between the groups occur. Then by taking out the common factor one would get a factorization of the whole expression



Example 7 Factor by grouping

2)a y y by b 1 1y y b y

1y y b

2 2)b x a b bx a 2 2x a bx b a

2x a b a b

2 1a b x

1 1a b x x



2) 4 13 10c x x 2 54 108x xx

4 2 5 2x x x

2 4 5x x

Example 8

Facto and write each of the following expressions without negative exponents

2 2 1) 27a x y x y 3/ 2 1/ 22) 2 1 2 1b a b a



Solution

2 2 1) 27a x y x y 1 2 3 327x y x y

2 2

2

3 3 9x y x xy y

xy

3/ 2 1/ 22) 2 1 2 1b a b a 3/ 2 222 1 2 1a b a


Factoring of a General Trinomial


2 ax bx c

Sometimes the factors of 2ax bx c can be

obtained by trial and error following the strategy

a. Find two numbers p and q such that pq = ac

and p + q = b.

b. 2 2 2 b p qax x c ax x c ax x qxp c c. Factor the last statement by grouping.



Example 8

Factor each of the following trinomials:

2. 10 13 3 a x x 5 1 2 3 x x

2. 2 2 b x x1 17 1 17

22 2

x x

Documents

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals B.1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals