Review of Algebra
September 23, 2022
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Outline
Set of Real Numbers
Some Properties of Real Numbers
Exponents and Radicals
Algebraic Expressions
Factoring
Fractions
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Set of Real Numbers
First, we will introduce the sets, the set of integers, the set of rational
numbers and the set of real numbers.
A set is a collection of objects.
An object in a set is called an element of that set.
The numbers 1,2,3, and so on form the set of positive integers, which is
denoted by Z+ ;
Z+ = f1, 2, 3, ...g
The positive integers, together with 0 and the negative integers
1, 2, 3, ..., form the set of integers, which is denoted by Z;
Z = f..., 3, 2, 1, 0, 1, 2, 3, ...g.
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Set of Real Numbers
A rational number is a number that can be written as a quotient of two
integers. So, the set of rational numbers, which is denoted by Q is ;
Q = f qp : where p, q 2 Z with q 6= 0g
Numbers represented by nonterminating nonrepeating decimals are
irrational numbers.
Example
p
2 is an irrational number.
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Set of Real Numbers
Together, the rational numbers and the irrational numbers form the set of
real numbers which is denoted by R.
Real numbers can be represented by points on a line.
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Some Properties of Real Numbers
We will illustrate some properties of the real numbers. Let a, b, and c be
real numbers.
1. The Transitive Property of Equality.
If a = b and b = c then a = c
2. The Closure Properties of Addition and Multiplication
For all real numbers a and b there are unique real numbers a + b and
ab.
3. The Commutative Properties of Addition and Multiplication
a + b = b + a and ab = ba
4. The Associative Properties of Addition and Multiplication
(a + b ) + c = a + (b + c ) and (ab )c = a(bc )
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Some Properties of Real Numbers
5. The identity properties.
There are unique real numbers denoted by 0 and 1 such that, for each
real number a,
0 + a = a and 1a = a
6. The Inverse Properties
For each real number a, there is a unique real number denoted by
such that
a + ( a) = 0
The number a is called the negative of a.
For each real number a, except 0, there is a unique real number
denoted by a 1 such that
a.a 1 = 1
The number a 1 is called the inverse of a.
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Some Properties of Real Numbers
7. The Distributive Properties
a(b + c ) = ab + ac and (b + c )a = ba + ca
Subtraction is de…ned in terms of addition:
b means a + ( b ) where ( b ) is the negative of b.
a
Division is de…ned in terms of multiplication. If b 6= 0, then
a
means a.(b
b
()
1)
where b
1
Review of Algebra
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Exponents and Radicals
In this section we will review positive exponents, the zero exponent,
negative exponents, rational exponents, principal roots, radicals and the
procedure of rationalizing the denominator.
Let x be any real number and n be a positive integer.
1
x n = x.x.x...x
| {z }
n factors
2
x
n
=
1
1
=
n
x
x.x.x...x
| {z }
n factors
3
4
1
= x n for x 6= 0.
x n
x0 = 1
The letter n in x n is called the exponent and x is called the base.
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Exponents and Radicals
If r n = x,
where n is a positive integer, then r is an nth root of x. The
p
n
symbol x is called a radical. Some numbers do not have an nth root
that is a real number. Thus;
p
positive if x is positive
n
x is
negative if x is negative and n is odd
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Properties of Exponents and Radicals
1. x m .x n = x m +n
2. x 0 = 1
3. x
n
=
1
xn
1
= xn
x n
xm
5. n = x m n
x
xm
6. m = 1
x
7. (x m )n = x mn
4.
8. (xy )n = x n y n
x n
xn
9.
= n
y
y
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Properties of Exponents and Radicals
10.
x
y
1
11. x n =
n
=
p
n
y
x
n
x
1
n
1
12. x
= p
n x
p p
p
13. n x n y = n xy
p
r
n
x
x
14. p
= n
n y
y
p
p
p
m n
15.
x = mn x
p
m
16. x n = n x m
p
17. ( m x )m = x
p
x if n is odd
18. n x n =
jx j if n is even
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Rationalizing Denominators
Rationalizing the denominator of a fraction is a procedure in which a
fraction having a radical in its denominator is expressed as an equal
fraction without a radical in its denominator.
Example
3
2
p
3
y2
=
2y 3
2
3y 3
()
p
1
23y
2y 3
=
=
2 1 =
3y 1
3y
3y 3 y 3
1
2
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Algebraic Expressions
If numbers, represented by symbols, are combined by any or all of the
operations of addition, subtraction multiplication, division, exponentiation,
and extraction of roots, then the resulting expression is called an algebraic
expression.
Examples
r
3x 2 2x + 5
is an algebraic expression in the variable x.
5 3x
(x + 2y )2 xy
2 is an algebraic expression in the variables x and y .
5y
3
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Algebraic Expressions
A polynomial in x is an algebraic expression of the form
cn x n + cn
1x
n 1
+ ... + c1 x + c0
where n is a nonnegative integer and the coe¢ cients c0 , c1 , ..., cn are
constants with cn 6= 0. The number n is called the degree of the
polynomial.
Example
x 3 + 5x 2 + x + 7 is a polynomial of degree 3.
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Algebraic Expressions
We can add, subtract, and group algebraic expressions term by term with
the same powers.
Example
5(x 2 y 2 ) + x (y 3x ) 4y (2x + 7y ) =
5x 2 5y 2 + xy 3x 2 8xy 28y 2 = 2x 2
33y 2
7xy
Example
4f3(t + 5) t [1 (t + 1)]g = 4f3t + 15 t [1 t 1]g =
4f3t + 15 t [ t ]g = 4f3t + 15 + t 2 g = 4t 2 + 12t + 60.
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Algebraic Expressions
We will give the list of special products that may be obtained from the
distributive property and are useful in multiplying algebraic expressions.
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Algebraic Expressions
Example
(x
2)(x + 3) = x 2 + ( 2 + 3)x + ( 2)(3) = x 2 + x
6
Example
( 3y + 2)(5y 4) = ( 3)(5)y 2 + (( 3)( 4) + (2.5))y + 2.( 4) =
15y 2 + 22y 8
Example
(z
4)3 = z 3
()
3.4.z 2 + 3.42 .z
43 = z 3
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12z 2 + 48z
64
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Long Division
To divide a polynomial by a polynomial, we use so called long division when
the degree of the divisor is less than or equal to the degree of the dividend.
dividend
remainder
=quotient+
divisor
divisor
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Factoring
In this part, we will state the basic rules for factoring and then apply them
to factor algebraic expressions.
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Factoring
Example
5r 2 st 2 + 10r 2 s 2 t 2 15r 2 t 2 = 5r 2 t 2 (s + 2s 2
5r 2 t 2 (2s + 3)(s 1)
3) = 5r 2 t 2 (2s 2 + s
3) =
Example
4x 2
9y 2 = (2x )2
(3y )2 = (2x
3y )(2x + 3y )
Example
27 + 8x 3 = (3)3 + (2x )3 = (3 + 2x )(9
()
6x + 4x 2 )
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Fractions
In this part, we will analyse how to simplify, add, subtract, multiply, and
divide algebraic fractions.
Simplifying Fractions
Completely factor both numerator and denominator.
Divide both numerator and denominator by the common factors.
Example
Simplify
x2
3x
x2
()
10
4
=
(x
(x
5)(x + 2)
(x
=
2)(x + 2)
(x
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5)
2)
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Fractions
Multiplication and Division of Fractions
c
a
The rule for multiplying by is
b
d
a c
ac
. =
b d
bd
The rule for divding
a
c
by where b 6= 0, d 6= 0, and c 6= 0 is
b
d
a
b
()
c
a d
ad
= . =
d
b c
bc
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Fractions
Example
Perform the operations and then simplify
2x 2
x2 1
2x 2
x 2 3x 4
=
.
=
x 2 2x 8 x 2 3x 4
x 2 2x 8
x2 1
2(x 1)(x + 1)(x 4)
2
(2x 2)(x 2 3x 4)
=
=
(x 2 2x 8)(x 2 1)
(x 4)(x + 2)(x 1)(x + 1)
(x + 2)
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Fractions
Adding and Subtracting of Fractions
If two fractions have common denominators then we can add and subtract
b
a
two fractions and as follows:
c
c
a b
a+b
+ =
c
c
c
a b
a b
=
c
c
c
Example
Perform the operations and then simplify
x2
5x + 6
x 2 + 5x + 6
(x + 3)(x + 2)
+
=
=
= x +2
x +3
x +3
x +3
x +3
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Fractions
Adding and Subtracting of Fractions
To add (or subtract) two fractions with di¤erent denominators we will
rewrite the fractions as fractions that have the same denominator. After
factoring all the denominators, we will …nd the least common denominator
of the fractions. Then we will convert the fractions into equal fractions
with least common denominator of the fractions.
Example
1
1
1
1
+
=
+
x 2 2x 3 x 2 9
(x 3)(x + 1) (x 3)(x + 3)
(x 3)(x + 1)(x + 3) is the least common denominator of the fractions,
thus we have
1
1
(x + 3)
+
=
+
(x 3)(x + 1) (x 3)(x + 3)
(x 3)(x + 1)(x + 3)
(x + 1)
(x + 3) + (x + 1)
=
=
(x 3)(x + 3)(x + 1)
(x 3)(x + 3)(x + 1)
(2x + 4)
2(x + 2)
=
(x 3)(x +
3)(x + 1)
(x 3Review
)(x of+Algebra
3)(x + 1)
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