Math Review for Algebra and Precalculus Stanley Ocken Department of Mathematics The City College of CUNY Copyright © January 2007 by Stanley Ocken No part of this document may be copied or reproduced[.]
Trang 1Math Review for Algebra and
Precalculus
Stanley Ocken Department of Mathematics The City College of CUNY
Trang 2Math Review for Algebra and Precalculus
Stanley Ocken Department of Mathematics The City College of CUNY
Copyright © January 2007
Trang 3Table of Contents Part I: Algebra Notes for Math 195
Introduction……… 3
1 Basic algebra laws; order of operations……… 4
2 How algebra works……….……….… 12
3 Simplifying polynomial expressions………29
4 Functions……… 41
5 When to use parentheses……… 55
6 Working with fractions……… 61
7 Adding fractions……… 76
Trang 4Math Review for Precalculus and Calculus
Part I: Algebra
Introduction
Algebra is the language of calculus, and calculus is needed for science and engineering When you attack a real-world problem, you want to represent the problem using algebra expressions When you read technical books, you want to be comfortable deciphering and working with these expressions Computers can’t do either of these tasks for you
Algebra used in undergraduate mathematics involves three main activities: rewriting expressions, solving equations, and solving inequalities You need to perform these somewhat mechanical activities quickly and accurately It’s very difficult to achieve this goal unless you understand how algebra works
Algebra is a symbolic language that allows communication between people who don’t know each others’ spoken language The grammar of the language involves three main components: expressions, identities, and equations
An expression involves numbers, variables, parentheses, and algebra operations Basic
types of expressions are integers, variables, monomials, polynomials, and so forth We’ll deal mostly with expressions in one variable, such as the polynomial x3− x+4.
An identity between two expressions, written with an equals sign, is a statement that each
expression can be obtained by rewriting the other A simple example isx+2=1+x+1 With rare exceptions, substituting numbers for variables turns an identity into a true
statement about numbers For example, setting x to 4 yields 4+2=1+4+1 An
important part of algebra is using identities to rewrite expressions
An equation is also a statement that two expressions are equal In most equations,
however, equality holds only for specific values of the variable For example, the
statement is true only when x is 1 or – 1 We say that the solutions of the
equation are x = 1 and x = – 1 Please remember;: we rewrite expressions but
we solve equations
3 2
x
3 2
x
In this preliminary edition, section headings such as AN1 are used for Algebra Notes, Section 1
Trang 5C HAPTER 1: B ASIC ALGEBRA LAWS ; ORDER OF OPERATIONS
1.1 Algebra operations and notation
Let’s begin with two tricky examples
Example 1.1.1: Rewrite 2x−1 as an expression with no negative powers
Right:
x x
⎠
⎞
⎜
⎝
⎛
=
−
Wrong:
x
x
2
1
Example 1.1.2: Simplify the expression −52
Right: −52 =−(52)=−25 Wrong: −52 =(−5)2 =25
To understand what’s going on, we need to review in some detail the five algebra
operations: addition, subtraction, multiplication, division, and exponentiation Each of
these is called a binary operation because it is used to combine two expressions
The table below lists notation and terminology for these operations The last entry shows
a special operation called negation, which operates on one expression and is an
abbreviation for multiplication by –1
Operation Write Say Describe the answer
Addition 4+ 3 4 plus 3 The sum of 4 and 3
Subtraction 4 – 3 4 subtract 3 or
4 minus 3
The difference of 4 and
3 Multiplication 4⋅ Traditional notation 3
3
4× Traditional notation 3
*
4 Calculator notation
) 3 (
4 +x Implied times sign Implied times sign
xy
4
4 times 3 The product of 4 and 3
Division 4÷3 Seldom used
3 /
4 Calculator notation
3
4
Traditional notation
4 divided by 3 four thirds
4 over 3 (slang)
The quotient of 4 by 3
Exponentiation 3
4 Traditional notation 4^3 Calculator notation
4 (raised) to the 3 rd (power)
The 3rd power of 4
4 is the base
3 is the exponent
Negation – 3
– (–5) = 5
Negative 3 or Minus 3 Negative of minus 5
The negation (additive opposite) of 3 is –3
It’s a bit annoying that the minus sign ‘–’ is used for three different purposes: naming a
negative number, subtraction, and negation Specifically:
Trang 6Example 7.6.2 Rewrite
x x x x
1
1
−
+
as a reduced fraction
Solution: Multiply both numerator and denominator by x
1
1 1
) (
1 ) ( 1 )
(
1 )
(
1
1
2
2
−
+
=
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛ +
=
⎟
⎠
⎞
⎜
⎝
⎛ −
⎟
⎠
⎞
⎜
⎝
⎛ +
=
−
+
x x x x x x
x x x x
x x
x
x x
x
x
x
x
x
The fraction doesn’t reduce, since x2+1 doesn’t factor
Example 7.6.3 Rewrite
1
1 1
1
1 1
− + + +
x
x as a reduced fraction
Solution: Multiply top and bottom by the LCD of the fractions
1
1 +
1
−
x In this
case the LCD is the product (x+1)(x−1) Then
x
x
x
x
x
x
x
x
x x
x
x x
x
x x x x
x
x x x x
x
x x
x
x x
x
)
1
(
) 2
)(
1
(
] 1 ) 1 )[(
1
(
] 1 ) 1 )[(
1
(
cancel can you if see to bottom and
top Factor the out!
multiply
t Don' ) 1 ( ) 1
)(
1
)(
1
(
) 1 ( ) 1
)(
1
)(
1
(
1
1 ) 1 )(
1 ( ) 1 )(
1
)(
1
(
1
1 ) 1 )(
1 ( ) 1 )(
1
)(
1
(
1
1 1 )
1
)(
1
(
1
1 1 )
1
)(
1
(
+
+
−
=
+
−
+
+ +
−
=
+ +
−
+
− +
−
+
⎟
⎠
⎞
⎜
⎝
⎛
−
− + +
−
+
⎟
⎠
⎞
⎜
⎝
⎛ +
− + +
−
+
=
⎟
⎠
⎞
⎜
⎝
⎛
− +
−
+
⎟
⎠
⎞
⎜
⎝
⎛
+ +
−
+
Trang 7Exercise 7.6.1 Rewrite each nested fraction as a fraction in standard form
a)
y
x
y
x
1
2
2
1
+
+
b)
4
1
3
1
3
2
2
1
+
+
c)
2
1
2
2
2
2
2
1
−
+
+
−
+
+
x
x
x
x
d)
d
c
b
a
4
3
2
1
+
+
e)
1
1
1
1
1
1
1
1
2
2
+
+
−
−
+
−
x
x
x
x
f)
9
1
6 2
1
3
1
−
+
+
x
x
x
g)
2
2
1
1
ba
b
ab
a
+
+