Faculty of Information Science & Technology
(FIST)
PAM 0135
Algebra
Foundation in Life Science
Foundation in Information Technology
ONLINE NOTES
Topic 1
Fundamentals of Algebra
FIST , MULTIMEDIA UNIVERSITY (436821-T)
MELAKA CAMPUS, JALAN AYER KEROH LAMA, 75450 MELAKA, MALAYSIA.
URL: http://fosee.mmu.edu.my/~asd/
PAM0135
Algebra
Topic 1
TOPIC 1:
FUNDAMENTALS OF ALGEBRA
Reference:
Michael Sullivan, et.al (2011). Algebra and Trigonometry. Pearson, Prentice Hall.
Objectives:
1.
Know the classification of numbers.
2.
Graph inequalities, find distance on the real number line, use the Laws of
Exponents and evaluate square roots.
3.
Add and subtract polynomials, multiply polynomials, know formulas for special
products and divide polynomials using long division.
4.
Factor the difference of two squares and the sum and the difference
of two cubes, factor perfect squares, factor a second-degree polynomial and
factor by grouping.
5.
Divide polynomials using synthetic division.
6.
Reduce a rational expression to lowest terms, multiply and divide rational
expressions, add and subtract rational expressions and simplify complex rational
expressions.
7.
Work with nth roots, simplify radicals, rationalize denominators and simplify
expressions with rational exponents
Contents:
R.1
Real Numbers
R.2
Algebra Essentials
R.3
Polynomials
R.4
Factoring Polynomials
R.5
Synthetic Division
R.6
Rational Expressions
R.7
nth Roots; Rational Exponents
R.1
REAL NUMBERS
Classification of Numbers
1.
Natural Numbers (N)
{1, 2, 3, . . .}
2.
Whole Numbers (W)
{0, 1, 2, . . .}
3.
Integers (Z)
Consist of:
{. . ., -3, -2, -1, 0, 1, 2, 3, . . .}
- positive integers
{1, 2, 3, . . . }
- negative integers
{ . . , -3, -2, -1}
- zero
{0}
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Algebra
Topic 1
4.
Rational Numbers (Q) - numbers that can be expressed as a quotient of two
integers.

a
Q
a and b are integers, b  0 
b

- may be represented as decimals: terminating, or non-terminating with repeating
digits.
3
3
2,
 0.375 ,
 0.272727...  0.27
Example:
8
11
5.
Irrational Numbers - represented by decimals that neither repeats nor terminates.
2  1.414213...,   3.14159...
Example:
6.
Real Numbers (R ) - all rational numbers together with all irrational numbers
Equality
1.
The reflexive Property:
aa
2.
The symmetric property:
If
a  b , then
ba.
3.
The principle of substitution: If a  b , we may substitute b for a in any
expressions with a.
Properties of Real Numbers
For all the following properties, a, b, c  R
终结
a  b is a real number; ab is a real number.
1.
Closure
:
2.
Commutative 交换Properties:
3.
Associative联合 Properties: a  b  c  a  b  c  a  b  c
a  b  c  a  b  c  a  b  c
4.
Distributive Properties:
5.
Identity 一致Properties:
0  a  a  0  a ; a 1  1 a  a
0 - additive identity; 1 - multiplicative identity.
6.
Additive Inverse Property:
a  (a)  a  a  0 ;
7.
Multiplicative Inverse Property:
a  b  b  a; a  b  b  a
a  b  c  a  b  a  c
a  b  c  a  c  b  c
a
1 1
  a  1; if
a a
(-a) - negative of a.
a  0;
1
- reciprocal
a
of a.
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Algebra
Topic 1
8.
Difference :
a  b  a  (b)
9.
Quotient:
a
1
 a  ; if
b
b
10.
Multiplication by zero:
a00
11.
Division Properties:
0
 0;
a
12.
Rules of Signs:
a(b)  (ab) ; (a)b  (ab) ; (a)(b)  ab
a
a
a
a a
 (a)  a ;

 ;

b
b
b
b b
13.
Cancellation Properties:
14.
Product law:
ab  bc implies a  c if b  0
ac a

if b  0 , c  0
bc b
If ab  0 , then a = 0 or b = 0 or both.
15.
Arithmetic of Quotients:
b0
a
 1; if
a
a0
a c ad  bc
 
if b  0, d  0
b d
bd
a c ac
 
if b  0, d  0
b d bd
a c a d ad
   
if b  0, c  0, d  0
b d b c bc
----------------------------------------------------------------------------------------------------------R.2
ALGEBRA ESSENTIALS
The Real Number Line
- Every real number corresponds to point on the line and, conversely, each point on the
line has a unique real number associated with it.
- The number line divides the real numbers into three classes: Negative real numbers,
Zero, and Positive real numbers.
Constants - a fixed number; a letter that represents a fixed number.
Variables - a letter used to represent any number from a given set of numbers.
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Algebra
Topic 1
Inequalities
< less than;
or equal to
ab
a0
a0
a0
a0
or




>

greater than;
less than or equal to; 
greater than
b  a  b  a is positive.
a is positive
a is negative
a is nonnegative
a is nonpositive
Absolute Value
if a  0
a
a 
if a  0
 a
a represents the distance to the origin from the point a.
Absolute value is defined by the rule:
Properties :
a, b  R
a  0,
a  a ,
a  0 a  0,
a a
 ,
b b
ab  a  b
Distance Between P & Q
If P and Q are 2 points on a real number line with coordinates a and b, respectively.
The distance between P & Q, is
d ( P, Q)  b  a  a  b  d Q, P
Exponents
a n  a  a  a  ...  a (n factors)
where a is a real number (base); n is a positive integer (exponent/power).
1
Note: a n  n ;
a 0  1, if a  0
a
Laws of Exponent:
1.
a n  a m  a mn
4.
am
 a mn
n
a
7.
a0  1
2.
a 
5.
an
a
   n
b
b
m n
 a mn
3.
a  b n
6.
a
 
b
n
n
 an  bn
a
 
b



n
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Algebra
Topic 1
Square Roots
b2  a
 b a
where a, b are nonnegative real numbers.
b is the principal square root of a.
a2  a
Note:
Properties:
1.
ab  a b
a

b
2.
a
b
Example:
5 10  50  25 2  5 2
1.
24
2.
3
 8 4 22 2
----------------------------------------------------------------------------------------------------------R.3
POLYNOMIALS
Monomial and Polynomial.
Monomial
A monomial in one variable is the product
of a constant times a variable raised to a
nonnegative integer power, axk where
Polynomial
A Polynomial is an algebraic expression of the
a = a constant known as a coefficient of the
monomial
where
x = a variable
k = an integer greater than zero known as
the degree of the monomial if
Example :
a0
form
an x n  an 1 x n 1  ....a1 x  a0
an , an1 ,....a1 , a0
= constants called the
coefficient of the polynomials
n  0 is an integer and known as the degree of
the polynomial if an  0
( a n is called the leading coefficient)
6x 2
 5x8
14
x 11
Example :
 6 x 3  5 x 2  14 x 
26 x13 
1
2
3 8
x  11x 5  7 x  9
5
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Algebra
Topic 1
Definition of the terms

Like terms - Two monomials with the same variable raised to the same power.

Binomial – The sum or difference of two monomials having different degree

Trinomial- The sum or difference of three monomials having three different
degree
Example : Identify the following expression either a monomial or polynomial or neither.
Hence, determine its coefficient and degree.
No
1
2
3
4
5
Expression
5x  2 x  3 x  5
5x7
8
2z3
5z 4 
 3z  
3
3
7 y6
3x  5
x2
3
Type
Coefficient
Degree
2
Adding and Subtracting Polynomials
Find the sum and difference of the polynomials x 5  12 x 3  8 x 2  7 and
 7 x8  3x5  4 x 4  6 x3  8 x  12
Using Horizontal Method.
x 5  12 x 3  8 x 2  7 - (  7 x8  3x5  4 x 4  6 x3  8 x  12 ) =
Using Vertical Method
x 5  0 x 4  12 x 3  8 x 2  0 x  7
minus
8
5
 7 x  3x  4 x 4  6 x3  0 x 2  8 x  12
=
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Algebra
Topic 1
Multiplying Polynomials
2
Find the product of 3x2  5 x  3x  8



Using Horizontal Method.
3x
2


 5 x  3x  8  3x 4  9 x 3  24 x 2  5 x 2  15 x  40
2
 3x 4  9 x 3  21x 2  15 x  40
Using Vertical Method
x
3x
2
2
5

 3x  8

=
Special Products
FOIL Method (First Outer Inner Last)
(ax+b)(cx+d) = ax(cx+d)+b(cx+d)
= axcx+adx+bcx+bd
=acx2+ (ad+bc)x+bd
Example : (2x+1) (3x-5) =
Difference of Two Squares.
(x-a)(x+a)=x2-a2
Example : (4x-3)(4x+3) =
Squares of Binomials or Perfect Squares.
(x+a)2=x2+2ax+a2
(x-a)2=x2-2ax+a2
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Algebra
Topic 1
Example :
a) (x+7)2 =
b) (x-5)2 =
Cubes of Binomials, or Perfect Cubes.
(x-a)3=x3-3ax2+3a2x-a3
(x+a)3=x3+3ax2+3a2x+a3
Example :
a) (x+2)3 =
b) (x-5)3 =
Difference and Sum of Two Cubes
(x-a)(x2+ax+a2) = x3-a3
(x+a)(x2-ax+a2) = x3+ a3
Example : (x+2)(x2-2x+4) =
Divide Polynomials Using Long Division
To divide two polynomials, we first must write each polynomial in standard form. To
check the answer obtained in a division problem, multiply the quotient by
the divisor and add the remainder.The answer should be the dividend.
(Quotient)(Divisor)+Remainder = Dividend
Example:
Find the quotient and the remainder when 3x 3  4 x 2  x  7 is divided by x 2  1
Solution:
Each polynomial is in standard form. The dividend is 3x 3  4 x 2  x  7 , and the
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Algebra
Topic 1
divisor is x 2  1
STEP 1: Divide the leading term of the dividend, 3x 3 by the leading term of the
divisor, x 2 . Enter the result, 3x, over the term 3x 3 , as follows:
3x
x2  1 3x3  4x2  x  7
STEP 2: Multiply 3x by x 2  1 and enter the result below the dividend.
3x
x2  1 3x3  4x2  x  7
3x 3
 3x
STEP 3: Subtract and bring down the remaining terms.
3x
x2  1 3x3  4x2  x  7
3x 3
 3x
4x2  2x  7
STEP 4: Repeat Steps 1–3 using 4x2  2x  7 as the dividend.
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Algebra
Topic 1
3x 4
x2  1 3x3  4x2  x  7
3x 3
 3x
4x2  2x  7
4x 2
4
 2x  3
Since x2 does not divide  2 evenly (that is, the result is not a monomial),
the process ends. The quotient is 3x  4 and the remainder is  2x  3
Check:
(Quotient)(Divisor)+Remainder


 (3x  4) x 2  1   2 x  3
 3x 3  3x  4 x 2  4   2 x  3
 3x 3  4 x 2  x  7 = Dividend
Then
 2x  3
3x3  4 x 2  x  7
3
x

4

=
x2  1
x2 1
----------------------------------------------------------------------------------------------------------R.4
FACTORING POLYNOMIALS
Common Factors
If a polynomial has a common factor, the common factor of highest degree is factored
out.
Example:
2a 2 x  4ax  6a  2a(ax)  2a(2 x)  2a(3)
{ the greatest common factor is 2a.}
 2a(ax  2 x  3)
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Algebra
Topic 1
Factoring Formulas
Formula
a. Difference of two squares :
a 2  b 2  (a  b)( a  b)
Examples
x 2  16  ( x  4)( x  4)
b.
x 2  8 x  16  ( x  4) 2
Perfect Square Trinomial:
2
2
a  2ab  b  (a  b)
a  2ab  b  (a  b)
2
2
2
4 x 2  4 xy  y 2  (2 x) 2  2(2 x)( y )  y 2
2
 (2 x  y ) 2
c. Sum of Two cubes
a 3  b3  (a  b)( a 2  ab  b 2 )
z 3  27  z 3  (3)3
d. Difference of two cubes
a 3  b3  (a  b)( a 2  ab  b 2 )
8 x 3  y 6  ( 2 x )3  ( y 2 )3
 ( z  3)( z 2  3z  9)
 (2 x  y 2 )( 4 x 2  2 xy2  y 4 )
Trial-and-Error Factorization
The factors of the second-degree polynomial
integers, have the form
(ax+b)(cx+d)
where
ac=p, ad+bc=q, and bd=r.
px 2  qx  r , where p,q, and r are
We use trial-and-error method to factor polynomials having this form since there is a
limited number of choices are possible.
Example:
x 2  2 x  3  ( x  1)( x  3)
Factoring by Regrouping
Sometimes a polynomials may be factored by regrouping and rearranging terms so that a
common term can be factored out.


x 3  x  x 2  1  x 3  x 2   x  1
Example:
 x 2 ( x  1)  1 x  1
 ( x  1)( x 2  1)
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R.5
Algebra
Topic 1
SYNTHETIC DIVISION
Synthetic Division
A Synthetic Division is a shortened version of long division for a polynomial.
We can use this method to
i) find the quotient and remainder
ii) verify a factor
iii) find the value of a polynomial
Examples:
1)
Find the quotient and remainder when
f (x) = 3x4 + 8x2 - 7x + 4 is divided by g(x) = x - 1
Solution:
The divisor is x -1, so row 3 entries will be multiplied by 1, entered in row 2 and added to
row 1.
13 0 8
7
4
row 1
3
3
11 4
row 2 (add)
--------------------------3 3
11
4
8
row 3
(quotient)
(remainder)
Then the quotient = 3x3 + 3x2 +11x + 4, and remainder = 8
Note: The quotient has degree 1 less than that of the dividend.
2)
Use synthetic division to show that g (x) = x + 3 is a factor of
f (x) = 2x5 + 5x4 - 2x3 + 2x2 -2x +3.
Solution:
The divisor is x + 3 = x - (-3)
3 2 5
2
2
 2 3 row 1
Then
-6
3 -3
3
-3
------------------------------------2 -1
1 -1
1
0
row 2 (add)
row 3
Remainder = 0  f (-3) = 0, then by factor theorem
g (x) = x + 3 is a factor of f (x).
3) Use synthetic division to find the value of f(x) = -3x4 + 2x3 - x + 1 at x = -2. ie. f (-2)
Solution:
Remainder Theorem says that the value of the polynomial function at x = -2 equals the
remainder when the function is divided by x - (-2).
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Algebra
Topic 1
Then,
2 3
2
0
1
1
6 -16
32 -62
-----------------------------3
8 -16
31 -61
Then, by Remainder Theorem, f (-2) = - 61
Note that there are 3 ways to find the value of a polynomial function f (x) at a number c:
1)
Substitution - replace x by the number c to find f (c).
2)
Remainder Theorem. - use synthetic division to divide f (x) by x - c.
The remainder is f (c).
3)
Use a calculator to find f(c).
----------------------------------------------------------------------------------------------------------R.6
RATIONAL EXPRESSIONS
Quotients of polynomials are called rational expressions.
Examples:
2x  1
4 x 2 y 5  2 xy
and
5x  3
5 xy  y
Simplifying Rational Expressions
A rational expression is simplified, or reduced to lowest terms, if the numerator and the
denominator have no common factors other than 1 and –1.
Example: The rational expression
x2  5x  6
( x  2)( x  3)
is first been factor then been simplified by canceling the common factor
( x + 3 ) and writing
x2  5x  6
( x  2)( x  3)

( x  2)( x  3) ( x  2)( x  3)
( x  2)

( x  2)
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Algebra
Topic 1
Multiplication and Division
The operations of multiplication and division are performed with rational expressions
in the same way they are with arithmetic fractions.
Example:
2 x ( x  1) 2 x( x  1)
1.


y ( y  1)
y ( y  1)
x  3 y2 1 x  3
x


 2
y
x
y
y 1
x( x  3)

y ( y 2  1)
2.
Addition and Subtraction
For rational expressions, the operations of addition and subtraction are performed by
finding a common denominator for the fractions and then adding or subtracting the
fractions.
Example:
2x  1 x  3

x  2 x 1
2x  1 x  1 x  3 x  2




x  2 x 1 x 1 x  2
(2 x  1)( x  1)  ( x  3)( x  2)

( x  2)( x  1)

[Common denominator is (x + 2)(x - 1)]
(2 x 2  3 x  1)  ( x 2  5 x  6)
( x  2)( x  1)
x 2  8x  5

( x  2)( x  1)
Complex Rational Expression
When sums and/or differences of rational expressions appear as the numerator
and/or denominator of a quotient, the quotient is called a complex rational expression.
x
y
Example : Simplify this complex rational expression:
x2
1 2
y
1
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Algebra
Topic 1
Solution:
x
y x

y
y y
 2
x2
y
x2
1 2

y
y2 y2
yx
y
 2
y  x2
y2
yx
 2
y  x2
y
y ( y  x)
= 2
y  x2
y ( y  x)

( y  x)( y  x)
y

yx
1
R.7
nth ROOTS; RATIONAL EXPONENTS
Radicals; nth Roots
n
a b
 a  bn ;
a is the radicand, n is the index.
Note: If n is even, a  0 , b  0 and
If n is odd, a, b  R .
n
an  a
Properties:
1.
3.
n
n
ab  n a n b
am 
 a
n
m
a
m
n
a

b
2.
n
4.
m n
2.
3
n
a
n
b
a  mn a
Example:
1.
3
8 x 4  2 x3 x
 16 x 4 y 7  2 xy 2 3 2 xy
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Algebra
Topic 1
Rationalize Denominator
To rationalize the denominator is to rewrite the quotient so that the denominator contains
no square roots.
Example:
5
5
2 5 2
1.



4
2 2 2 2 2
2
2
3 x 2 3 x
2.



3 x
3 x
3 x 3 x


Simplify Expressions with Rational Exponents
Radicals are used to define rational exponents.
1
n
If a is a real number and n  2 is an integer, then a  n a , provided that
Note that if n is even and a  0 then
n
n
a exists.
1
n
a and a do not exist.
If a is a real number and m and n are integers containing no common factors,
m
n
with n  2 , then a  n a m 
Example:
1. 32

2
5

 32 
2
5
3
2. 254  25 2 
6
n
m
, provided that
n
a exists.
1
4
 2 2 
 25   5
3
 a
3
 125
1
2
 
5
9 x 
 2 13 
 2  13 
 53 
 
6
 9x y 
 9x 
 9x 
   3x
3.  1   






1
2
1
1
2 2
 x3 y 
 y 1 3 
 y3 
3


y






 y3 
 
 
Example:
1
4
4 3
x 2 x  1  2 x 3
Factor:
3
1
2
1
2
5
3
1
2
1
2
Solution:
1
3
4
3
1
3


1
3
4
4 x 2 x  1 6 x
2 x 22 x  1  3x
2 x 7 x  2
x 2 x  1  2 x 



3
3
3
3
3
1
3
4
3
1
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PAM0135
Algebra
Topic 1
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