# Chapter 1 Linear Equations and Graphs

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```					Appendix A

Basic Algebra
Review

Highlights
Basic Real Number Properties

Let a, b, and c be arbitrary elements in the set of real numbers R.

Associative: (a + b) + c = a + (b + c)
Commutative: a + b = b + a
Identity: 0 is the additive identity; that is, 0 + a = a + 0 = a for all
a in R, and 0 is the only element in R with this property
Inverse: For each a in R, –a, is its unique additive inverse; that is,
a + (–a) = (–a) + a = 0 and –a is the only element in R
relative to a with this property.

Basic Real Number Properties

Let a, b, and c be arbitrary elements in the set of real numbers R.

Multiplication Properties
Associative: (ab)c = a(bc)
Commutative: ab = ba
Identity: 1 is the multiplicative identity; that is, (1)a = a(1) = a for
all a in R, and 1 is the only element in R with this
property.
Inverse: For each a in R, a  0, 1/a is its unique multiplicative
inverse; that is, a(1/a) = (1/a)a = 1 and 1/a is the only
element in R relative to a with this property.
Basic Real Number Properties

Distributive Properties

5(3 + 4) = 5 • 3 + 5 • 4 = 15 + 20 = 35

9(m + n) = 9m + 9n

(7 + 2)u = 7u + 2u

Further Properties

Negative Properties

For all real numbers a and b,

1.  a   a                                a   a a
5.          ,   b0
b    b b
2. a b   ab 
a    a    a a
3. a b   ab                       6.            ,     b0
b    b     b b
4. 1a  a

Further Properties

Zero Properties

For all real numbers a and b,

1. a  0  0

2. ab  0          if and only if          a0   or   b0

Fraction Properties

The quotient a ÷ b (b  0) written as a/b is called a fraction.
The quantity a is called the numerator, and the quantity b is
called the denominator.
For all real numbers a, b, c, d, and k, (division by 0 excluded)
a c
1.               if and only if         ad  bc
b d
ka a
2.     
kb b                      (Note: 3 and 4 are on slide 25.)
a c ac                           a c ac               a c ad  bc
5.                              6.                  7.  
b b  b                            b b  b                b d   bd
Natural Number Exponent

Natural Number Exponent
For n a natural number and b any real number,
b n  b  b ... b         n factors of b

where n is called the exponent and b is called the base.

First Property of Exponents
For any natural numbers m and n, and any real number b,

b m  b n  b mn

Polynomials

Algebraic expressions
Constants and variables and the algebraic operations of
addition, subtraction, multiplication, division, raising to
powers, and taking roots. Special types of algebraic
expressions are called polynomials.
Polynomial in one variable x
Adding or subtracting constants and terms of the form axn,
where a is a real number and n is a natural number.
Polynomial in two variables x and y
Adding or subtracting constants and terms of the form axmyn,
where a is a real number and m and n are a natural numbers.

Polynomials

Degree of the term
Power of the variable.
Degree of the term with two or more variables
Sum of the power of the variables.
Degree of the polynomial
Degree of the nonzero term with the highest degree in the
polynomial.
Polynomial of degree 0
A nonzero constant. The number 0 is also a polynomial but
is not assigned a degree.

Combining Like Terms

A constant in a term of a polynomial, including the
sign that precedes it, is called the numerical
coefficient, or simply, the coefficient, of the term.

Distributive Properties of Real Numbers
1. a b  c   b  c a  ab  ac
2. a b  c   b  c a  ab  ac
3. a b  c  ...  f   ab  ac  ...  af

Combining Like Terms

Two terms in a polynomial are called like terms if they have
exactly the same variable factors to the same powers.
If a polynomial contains two or more like terms, these terms
can be combined into a single term by making use of
distributive properties.

3x 2 y  5xy 2  x 2 y  2x 2 y  3x 2 y  x 2 y  2x 2 y  5xy 2
                  
 3x 2 y  x 2 y  2x 2 y  5xy 2
 3  1  5 x 2 y  5xy 2
 2x 2 y  5xy 2

Addition and subtraction of polynomials can be thought of in
terms of removing parentheses and combining like terms.

x 4  3x 3  x 2 ,  x 3  2x 2  3x, 3x 2  4x  5

x   4
                  
 3x 3  x 2  x 3  2x 2  3x  3x 2  4x  5   
 x 4  3x 3  x 2  x 3  2x 2  3x  3x 2  4 x  5
 x 4  4x 3  2x 2  x  5

Multiplication

Multiplication of algebraic expressions involves the extensive
use of distributive properties for real numbers, as well as
other real number properties.

Multiply:        2x  33x 2  2x  3

2x  33x 2  2x  3 2x 3x 2  2x  3 33x 2  2x  3
 6x 3  4 x 2  6x  9x 2  6x  9
 6x 3  13x 2  12x  9

Multiplication

Thus, to multiply two polynomials, multiply each term of one
by each term of the other, and combine like terms.

For the product of two binomials we use this process:

2x  13x  2   6x 2  4x  3x  2
 6x 2  x  2

Special Products

Special products:

1.    a  b a  b   a 2  b 2

2.    a  b   a 2  2ab  b 2
2

3.    a  b 2  a 2  2ab  b 2

Combined Operations

Note that in simplifying, we usually remove grouping symbols
starting from the inside. That is, we remove parentheses ( ) first,
then brackets [ ], and finally braces { }, if present.

Order of Operations
Multiplication and division precede addition and subtraction,
and taking powers precedes multiplication and division.
                                  
3x  5 3  x  x 3  x   3x  5  3  x  3x  x 2 
                                                    

 3x  5  3x  9x  3x 2   
 3x  5  3x  9x  3x 2
 3x 2  3x  5
Polynomial in Factored Form

A polynomial is written in factored form if it is written as
the product of two or more polynomials.

A polynomial with integer coefficients is said to be factored
completely if each factor cannot be expressed as the product
of two or more polynomials with integer coefficients, other
than itself or 1.

Common Factors

Generally, a first step in any factoring procedure is to factor
out all factors common to all terms.

Factor out all factors common to all terms.

A. 3x 3 y  6x 2 y 2  3xy 3  3xy x 2  3xy 2xy  3xy y 2

 3xy x 2  2xy  y 2   
B. 3y 2y  5   2 2y  5   3y 2y  5   2 2y  5 
 3y  2 2y  5 

Special Factoring Formulas

u  2uv  v  u  v 
2              2               2
Perfect square:
u 2  2uv  v 2  u  v 
2
Perfect square:
Difference of squares:                         u 2  v 2  u  v u  v 
Difference of cubes:                                                
u 3  v 3  u  v  u 2  uv  v 2    
Sum of cubes:                                  u 3  v3    u  v u   2
 uv  v 
2

Rational Expressions

A quotient of two algebraic expressions (division by 0
excluded) is called a fractional expression.

If both the numerator and the denominator are polynomials,
the fractional expression is called a rational expression.

1                  5              x7         x 2  2x  4
x 3  2x               x          3x 2  5x  1         1

Rational Expressions

AGREEMENT Variable Restriction
Even though not always explicitly stated, we always assume
that variables are restricted so that division by 0 is excluded.

For example, given the rational expression
2x  5
x x  2 x  3

the variable x is understood to be restricted from being 0, –2,
or 3, since these values would cause the denominator to be 0.

Reducing to Lowest Terms

Fundamental Property of Fractions
If a, b, and k are real numbers with b, k  0, then

ka a

kb b

Reducing to Lowest Terms

Using this property from left to right to eliminate all common
factors from the numerator and the denominator of a given
fraction is referred to as reducing a fraction to lowest terms.

Using the property from right to left–that is, multiplying the
numerator and denominator by the same nonzero factor–is
referred to as raising a fraction to higher terms.

Reduce to lowest terms.
x 4  8x


x x  2  x 2  2x  4   
3x  2x  8x
3       2
x x  2 3x  4 
x 2  2x  4

3x  4
Multiplication and Division

Multiplication and Division

If a, b, c, and d are real numbers, then

a c ac
1.    ,    b, d  0
b d bd
a c a d
2.      ,     b, c, d  0
b d b c

Multiplication and Division

Perform the indicated operation and reduce to lowest terms.
10x 3 y   4x 2  12x    10x 3 y   x2  9
                       2
3xy  9y    x 9
2
3xy  9y 4x  12x
10x 3 y x  3x  3
           
3y x  3 4x x  3
5 x2         11

10x y3
x  3 x  3
             
3 y x  3     4x x  3
31          21

5x 2

6

For a, b, and c real numbers,

a c ac
1.       ,               b0
b b  b
a c ac
2.       ,               b0
b b  b

Least Common Denominator

The least common denominator (LCD) of two or more
rational expressions is found as follows:
1. Factor each denominator completely, including integer
factors.
2. Identify each different factor from all the denominators.
3. Form a product using each different factor to the highest
power that occurs in any one denominator. This product is
the LCD.

Combine into a single fraction and reduce to lowest terms.
1  1    2                           1   1       2
  2                                
x 1 x x 1                         x  1 x x  1x  1
LCD = x(x – 1)(x + 1)                   x x  1  x  1x  1  2x

x x  1x  1
x 2  x  x 2  1  2x        1 x
                        
x x  1x  1      x x  1x  1
 x  1     1
                  
x x  1 x  1 x x  1

Compound Fractions

A fractional expression with fractions in its numerator,
denominator, or both is called a compound fraction. It is
often necessary to represent a compound fraction as a simple
fraction–that is (in all cases we will consider), as the
quotient of two polynomials.

We will use the two different methods.

Compound Fractions

Express as a simple fraction reduced to lowest terms. Use
division of rational forms.
1     1

5h 5   1  1 h
h       5  h 5 1
           
55h 1
           
5 5  h  h
h

5 5  h h
1

5 5  h 
Compound Fractions

Express as a simple fraction reduced to lowest terms. Multiply
the numerator and denominator by the LCD of all fractions.
y      x      2 2 y             x             y              x
         x y  2                   x 2 y2 2  x 2 y2
x 2
y 2
x              y2 
          x               y2
                          
y      x       2 2 y            x         2 2 y              x
          x y                      x y  x 2 y2
x      y          x             y
             x              y

y3  x 3
 3        

y  x  y 2  xy  x 2            
xy  x y      xy y  x  y  x 
3

y 2  xy  x 2                  x 2  xy  y 2
                        or
xy y  x                      xy x  y 
Integer Exponents

For n an integer and a a real number:

1. For n a positive integer,

a n  a  a ... a         n factors of a

2. For n = 0,

a0  1        a0
0 0 is not defined.

Integer Exponents

For n an integer and a a real number:

3. For n a negative integer,
1
a  n
n
a0
a
If n is negative, then (–n) is positive.

Note: It can be shown that for all integers n,
1                           1
a n             and        an           a0
an                         a n
Exponents Properties

For n and m integers and a and b real numbers,
1. a m a n  a mn

2.    a 
n m
 a mn

3.    ab   m
 a mb m
m
 a am
4.    m
 b b
am           1
5.   n
 a  nm
mn
a0
a          a
Exponents Properties

Simplify, and express the answers using positive exponents only.
(p. 565)
  
A. 2x 3 3x 5  2  3x 35  6x 8

x5        2 1
C.       7
x x  2
57

x            x
u6
E.    u v   u  v 
3 2 2          3 2        2 2
 u 6 v 4  4
v

4m 3n 5 2m 34  2m
G.   4 3
    3 5 
 8
6m n       3n           3n
Exponents Properties

Write as a simple fraction with positive exponents.
1 x                            1 x

x 1  1                        1
1
x
x 1  x 

1 
x   1
x 
x 1  x 

1 x
x
nth Roots of Real Numbers

For any natural number n,
r is the nth root of b if rn = b

Real nth root of b
b1 n        or     n
b

n
b
nth Roots of Real Numbers

Evaluate the following. (p. 569)
A. 41 2  4  2

B.  41 2   4  2

C.     4   12
 4 are not real numbers

D. 81 3  3 8  2

E.    8  3 8  2
13

F.  81 3   3 8  2
Rational Exponents

If m and n are natural numbers without common prime
factors, b is a real number, and b is nonnegative when n is
even, then
   
 b1 n m  n b m

b 
mn

b

m 1n
 
 n bm
and
m n     1
b       mn      b0
b
Note that the two definitions of bm/n are equivalent under
the indicated restrictions on m, n, and b.
Rational Exponents

Change rational exponent form to radical form. (p. 570)

A. x1 7  7 x

3u v                  3u v                      3u v 
3
2 3 35               2 3 3
B.                       5
or            5        2 3

2 3     1     1                            2                      1
C. y            23                 or    3
y            or         3
y     3 2
y                                                    y2

Rational Exponents

Change radical form to rational exponent form.

D.     5
6  61 5

E.  3 x 2  x 2 3

F.         x y  x y
2    2
   2      2 12


Rational Exponents

Multiply, and express answers using positive exponents only.

               
A. 3y 2 3 2y1 3  y 2  6y 2 31 3  3y 2 32
 6y  3y 8 3

B.     2u   12
                 
 v1 2 u1 2  3v1 2  2u  5u1 2 v1 2  3v

If c, n, and m are natural numbers greater than or equal to 2,
and if x and y are positive real numbers, then

1.     n
xn  x

2.     n    xy  n x n y

n
x          x
3.     n     
y      n
x

Simplify using properties of radicals. (p. 572)

A.    4
3x y 
4   3 4
 3x 4 y 3

B.    4
8 4 2  4 16  4 2 4  2

xy      3 xy 3 xy               13
C.     3                            or      xy
27      3
27   3                3

Rationalize each denominator.

6x    6x   2x 6x 2x
A.                       3x 2x
2x    2x 2x    2x

B.
6

6

7 5 6

 7  5  3 7  5 
7 5              7 5 7 5               2

C.
x4

x4

x  2 x  4  x  2
                x 2
     
x 2   x 2 x 2         x4