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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
B.1
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Polynomials
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Definitions Polynomial Evaluation Polynomial Equations Algebra of polynomials Factoring Polynomials
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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.
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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.
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4 23x y22 abc
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Note:
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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.
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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Polynomials in One Variable
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Polynomial Evaluation
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Algebra of Polynomials
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Scalar Multiplication
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Multiplication of Polynomials
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Special Products
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Factoring of Polynomials
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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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Factoring by Grouping
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Factoring of a General Trinomial
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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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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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