Download Math 1311 – Business Math I

Math 1311 – Business Math I
Day 1: Review
Sets of Numbers
____ = { 1, 2, 3, 4, ..... } is called the set of __________________________.
Let x be one of these numbers. Find the solution of the following equations.
2x + 4 = 6  x = _________
x + 2 = 2  x = ___________
_____ = { 0, 1, 2, 3, ... } is called the set of _________________________
Let x be one of these numbers (
) . Find the solution of the following equations.
x – 4 = 6  x = ___________
x+6 =2, 
x = ____________
______ = { ..., -3, -2, -1, 0, 1, 2, 3, ... } is called the set of __________________
Let x be one of these numbers. Find the solution of the following equations.
x – 4 = - 6  x = _____
2x + 1 = 2  x = _________
Other Sets and other names for the sets above
With respect to integers the set { 1, 2, 3, ... } is also called the set of __________________________________
With respect to integers the set { 0, 1, 2, 3, 4, .... } is also called the _____________
{ 2, 3, 5, 7, 11, ... } represents the set of ____________________________
If a natural number is not prime and is greater than 1, then it is called _____________
The set that contains all real numbers that can be written as fractions is called the set of __________________________
We describe this set in what is called set-builder notation: Q = { a / b | a is an integer and b is a nonzero integer}
Real numbers that are not rational numbers are called _________________________________
( this means the real numbers are made up entirely of rational and irrational numbers)
If a real number can not be written as a fraction – we call it an ______________________________
Rational: ________________________
Irrational:______________________________
Real numbers
Irrational #’s
Rational
pure fractions
integers
negative integers
nonnegative integers
(whole numbers)
zero
positive integers
(natural numbers, counting numbers)
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Properties of Real Numbers
commutative law of addition
a + b = _________
commutative law of multiplication
ab = ________
examples:
( 3 ) + ( - 4 ) = _________
associative law of addition
a + ( b + c ) = _________
( 2/3 ) • ( 5/7) = ____________
associative law of multiplication
( a b ) c = ____________
examples:
1.3 + ( 3.7 + 4.8) = _____________
(
3 •
2 )•
2
Distributive law of multiplication over addition
a(b + c ) = ________________
examples:
3 • ( -2 + x ) = __________
1.7 ( 2.9 – 1.2 ) = _______________
commutative and the associative laws do not hold true with subtraction and division – do they ?
ex.
2–4 = 4 -2 ?
________
ex. ( 12 – 4 ) - 2 = 12 - ( 4 – 2 ) ? _________
12  6 = 6  12 ? ____________
(16  8 )  2 = 16  ( 8  2 ) ? ________
Other terms
Given a natural number(greater than 1): the number is either prime or it is composite. If it is composite, it can be written as a
product of prime numbers ( prime factors). The product is called the prime factorization of the number.
ex. 24 = _____________
ex. 150 = ____________
ex. 95 = ____________
ex. 29 = ______________
2
GCF - greatest common factor
Given two or more natural numbers we can find the greatest ( largest) number that will divide evenly into all of the given
numbers. This is called the greatest common factor – GCF ( or sometimes: the GCD – greatest common divisor)
Find GCF of
12 and 20 ___________
( 8, 20, 30 ) = ___________
248 and 324
LCM – least common multiple
Given two or more natural numbers we can find the smallest (least) number that all the given
numbers will divide evenly into.
Find LCM ( 20, 15 ) = ________
LCM ( 2, 3, 5 )
Find GCF and LCM of 150 and 240. _______________
____________________
Absolute Value of a real number: Let x be a real number. The absolute value of x, written | x |, is defined by
x if x > 0
| x | = either
- x if x < 0
ex. | 6 | = _________
| - 2 | = _________
| x | = _______ if x is a natural
- | - 4 | = _________,
number
3
Inequalities
< : means less than
> : means greater than
True or False
4< -4
 : means less than or equal to
ex.
4 < 6  we say 4 is less than 6
 : means greater than or equal to
ex.
12  10  we say 12 is greater than or
equal to 10
-2  - 2
3/ 11 >
2/9
______________
______________
______________
Using inequalities we can represent numbers on a number line –
0.01 < 0.0091
________________
x  2
x< 4
-4 < x  2
Integer Exponents
xn : exponential notation, x is called the ___________ and n is called the ____________ or the ___________
and xn = xxxx … x ( a total of n x’s) – this is the expanded notation
ex.
25 = _____________________
( - 4) 3 = __________________
ex. – 3  3  3  3 = _____________
ex. – 42 = _____________
- ( 4)2 = _________
( 2/3 ) 4 = ___________
ex. 0.014 = ________
Def. If x is a real number not equal to zero, then xo = ___________
( 4)0 = _______
( - 2 ) 0 = _______ ,
What about - ( 4 ) 0 = _________
So,
( - c) 0 = _________ for c  0
- 3 0 = ___________
and - ( - 2 )0 = ______ .
( 0) 4 = ________
What is 0 0 = ______
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Def. If n is a natural number, then x – n = ________ . This gives us a way to work with negative exponents in terms of
natural numbers.
ex. Find
2-2 = ___________
What about
( ¼) – 2 = __________
4-3 = ____________
8-1 = __________
( ½) – 3 = _____________
( ¾)-2 = __________
Properties of exponents. Let x and y represent real numbers and let m and n be integers.
1. xn  xm = ____________
x4  x5 = ____________
x8  x = ___________
2. xn  xm = ______________
x 8  x4 = ___________
x12 / x20 = __________
3. (xn ) m = _______________
(x3 )2 = _________
(x6 )3 = _________
4. ( x  y ) n = _____________
(2x)3 = _____________
( 4xy)3 = ___________
( x2y3 )2 = _________
( 2xy3 )3 = _____________
( x/y)4 = __________
( x2 / y3 ) 3 = ____________
5. ( x  y ) n = ____________
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Other examples.
1) ( 2x3 y)  ( - 4x2y4 ) = __________________
4xy
----------12xy6
3)

2) ( 2xy2 )2  ( 3x2y )3 = _____________
- 2y3
------------- = _______________
x2y
-2x
3xy3
2
4) ( --------- )  ( ---------- )2
y2
-2y3
= _____________
Since we have a definition for negative exponents – how would we work problems with negative exponents.
1. ( x
-2 -4
)
= _______________
( x-3 )2 = ______________
2.
( x-3y-4)-2 = _______________
( 2x-2 )-2 = ________________
( 2x-3y0 )  ( 3x-1y ) = ________________
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Since the rules work with integral exponents – do they work with rational exponents ? Yes
1)
x 2/3  x7/3 = ___________
2) x 1/5 y  x y 2/3 = ___________
3) ( x -2/3 ) –6 = _____________
4) ( - 5x1/2y)1/2 = __________
5.
More Examples of exponent problems –
Find
( - 2xy3 ) 0 = ____________ if neither x nor y = 0
0 4 = ____________ 0
-2
= ____________
18 0 = ___________
00 = _________
- 40 = _____________
- 42 = ___________
- 4 – 2 = _____________
Also, if x = -1 and y = - 2 and z = 0 find
1) xy ____________
2) y0 = __________
3) - x2 = ____________ 4) (x – y )  x + y = ___________
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Radicals
square roots: of 49 ==> ________________ square roots of
Cube roots of 64 ==> ______________
81 ==> ________________
cube roots of - 27 ==> ________________
Continue by asking for 4th roots, 5th roots,….Notice that each number has either 1, 2, or none nth roots.
ex. Find the square roots of 25  ______________
Find the fifth roots of – 32  ___________
Find the 4th roots of ( -16)  ___________
n
Define Principal nth roots of x:
___
\/ x
n __
if x > 0, then \/ x > 0 
; n is called the _____________, x is the _____________
4 ____
\/ 125 = ____________
n __
3 ____
if x < 0 and n is odd, then \/ x
< 0  \/ - 27 = _________
n __
2 ___
if x < 0 and n is even, then \/ x has no real value  \/ -4 = ____________
ex. Find each of the following nth roots
___
3 ___
1) \/ 64 = _________
2) \/ - 8 = ____________
4 ___
4) \/ x8 = ___________
True or False:
2 ____
3 ) \/ - 25 = ____________
2 _____
5) \/ x6y4 = ___________
__
\/ x2 = x ___________________
So,
_____
\/ 8x3y12
3
6)
= __________
_____
\/ ( -2)2 = _________
However, if we assume that x is a positive real number, then
___
\/ x2 = x
8
Define x
1/n
n ___
= \/ x
Examples of x1/ n
1) 8
1/3
2) 16 ¼ = ___________________
= ____________________
3) - 25 ½ = __________________
5) ( - 9 )
½
4) ( - 27 )
= _______________
6) ( 16) – 1 / 4 = _________________
= ________________
n ___
Define x m/n = \/ x m
1/3
n __
m
or ( \/ x )
Now we can use fractional exponents: 16 ¾ = __________,
ex. (16x2/3y9 ) 1/ 4 = _____________
- 9 3/2 = ____________ , - 16-1/2 = ________
ex. ( -8x6y9 ) 2/3 = _______________
ex. ( 4x-4y8 ) – ¾ = _______________
More on radicals:
While we can simplify quantities that are perfect squares, perfect cubes,…
3 ___
4 _______
such as \/ -64 = _______ or \/ x8 y12
= __________
What about
___
\/ 20 = _________
__
\/ 16 = ________
3
___
\/ x8
3
9
Properties:
n ___
1) \/ xy
n
2)
__
n __
\/ x  \/ y
n
_______
x
\/ ----=
y
/
__
\/ x
n
=
___
\/ x
----------- 
\/
y
n
__
\/ y  ex.
n
____
\/ 4x2
= ______________
________
9x4
ex. / -------- = __________________
\/
4y2
/
ex. Write 42 in prime factored form ( as a product of prime numbers ) = ________________
ex. What is the prime factorization of 24 ? 24 = ______________
We say a radical is in simplest form if ( radical must be in prime factored form )
1) The index is smaller than all of the exponents inside the radical
5 ___
4 ______
ex. \/ x8 = _________
ex. \/ 16x5y2
= ______________
2) There is no common factor between the index and all of the exponents inside the radical
4 ____
6 ______
ex. \/ x2
= ________
\/ 8x3y3 = ____________
____
\/ 2x2 = __________
4
3) There can be no radical in the denominator or no denominator inside the radical.
ex.
2
-------- = __________
\/ 2
ex.
4x
--------- = ____________
\/ 4x
10
_______
9x
/ -------\/
4y
ex.
/
________
9x
-------3 /
\/
4y2
ex.
/
Other radical problems:
sum/difference:
__
Find
4 \/ 9 = ________
__
4 + 2 \/ 9
__
\/ 2 = ______________
3
__
2\/ 3
= _________
products / quotients.
__
___
\/ 5  \/ 20 = ___________
__
\/ 2
3 __
9 + \/ 8 = _______________
__
\/ 4x
___
- 3 \/ 12 = _____________
___
\/ 2x2 = ___________
______
___
\/ \/ 8 = _______________
3
/
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Polynomials
(factors, terms, degree)
Sum of literal expressions in which each term consists of a product of constants and variables with the restriction that each
variable must have a nonnegative integer exponent.
2x,
3x2y – 4,
1 + x + 3x2, 5 – 3xy + y9, ….  What about __________ or ____________ ?
Special types of polynomials
if one term:
__________
if two terms:
___________
if three terms:
__________
How many terms does each of the polynomials have ?
2xy
_______________
Def. (degree)
monomials :
3 + 2xy + x9 + y2
__________________________
1 + x + x2
__________________
3x ==> _________
2x5 ==> ___________
½ x5 y3 ==> ____________
binomials: 2x – 1 ==> ________
x + y ==> _________
3x 2y - 4x3y2 ==> ____________
other polynomials:
x – 2xy + y3 ==> ___________ x10 - 2xy + x6y5 ==> _________-
Basic operations of polynomials: sum – difference, products - quotients
Find ( sum and differences)
a) (2xy - 4x ) + ( 3xy - 2x ) ==> _____________
b) ( 3x2 - 2x + 3 ) - ( 2x2 - 4x - 5 ) ==> __________
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Products
a) 2(x – 3y ) = _______________
b) x( x + 2y ) = ___________________
c) 3x3 ( 3xy – 2x4y ) = ______________
d) ( 2x –3y )( 4x + 2y ) = _________________
Def. The process of writing a polynomial as a product of other polynomials of equal or lesser degree is called
__________________
Recall: GCF
Find GCF ( 20, 36 ) = ____________
GCF ( xy, x ) = __________
GCF(2xy, 6y2 ) = _________
GCF ( 12x3y6, 8x4y2 ) = ___________
Special Products and Factoring
1) greatest common factor:
x( y + x) = ________________
2) difference of squares : (x – y ) ( x + y ) = _____________
3) sum – difference of cubes : ( x – y ) ( x2 + xy + y2 ) = ___________
13
4) perfect squares: (x + y )2 = __________________ = ________________
5) Trinomials of the form ax2 + bx + c
Review of methods of factoring Factoring: process of writing a polynomial as a product of other polynomials of equal or lesser degree.
Methods:
GCF – always look for a common factor - 1st method
Difference of squares: x2 - y2 = (x – y ) ( x + y)
Sum – Difference of Cubes: x3 + y3 = ( x + y ) ( x2 - xy + y2 ) ,
x3 - y3 = ( x – y ) ( x2 + xy + y2 ) -- SOPPS
Perfect Squares: x2 + 2xy + y2, first and last must always be positive
Multiply (x + 2y )2 = _____________
Factor: x2 - 8x + 64 = _______________
x2 + 4xy + 4y2 = ___________
(3x – 4y )2 = _____________________
x2 + 10x + 25 = __________________
4x2 - 12xy + 9y2 = _________________
x2 - 16x - 16 = ___________
14
Grouping –
2 ( x – y ) - y ( x – y ) = _________________
xy + 2x - y2 - 2y = _______________
x2 - y2 – 2y - 4 = ______________________
x3 - y3 - x + y = _______________
More on Factoring:
GCF:
2 – 12x = ___________________
x2 + 2x = ___________________
x(y-1) – y( y – 1 ) = ________________
2x ( y - 4 ) +
3y( 4 – y ) = _______________
Difference of Squares:
x2 – 36 = _________________
x3 - 4x = _____________________
12x2y - 27y3 = _______________
x4 - 16 = ____________________
x2 + 9 = ________
Perfect Squares:
x2 + 12x + 36 = _____________________
x2 – 8x + 64 = __________________
x2 + 10x + 25 = _____________________
x2 – 4x - 4 = ____________
x3 – 6x2 + 9x
15
Sum/Difference of Cubes:
x3 - 8 = ________________________
x3 + 27 = ________________________
8x3 + 64 = ___________________________
1 - 8y3 = __________________________
Trinomials of the form ax2 + bx + c
Case I : a = 1, factor x2 + bx + c
x2 + 6x + 5 = ______________________
x2 + 8x + 7 = ________________
x2 - 8x +
x2 - 6x + 8 = ___________________
15 = ____________________
Case II: If a  0, factor ax2 + bx + c
2x2 + 5x + 2 = _______________________
3x2 - 8x + 4 = ____________________
4x2 - 4x - 3 = ________________________
4x2 + 2x - 6 = ____________________
Grouping:
2xy – 3x - 4y + 6 = _________________
16
Algebraic Fractions -- quotients of polynomials.
ex. 2x / ( x2 - 3x + 4 ) , ( x - 4 ) / 3, ( x2 + 1 ) / (x2 + 2x + 1), ¾, …
We can reduce these fractions to equivalent fractions with a smaller denominator and we can perform algebraic operations on
these fractions: add, subtract, multiply, divide, ….
ex. Reduce each of the following fractions to lowest terms 
x–1
a) -------- =
x2 - 1
2x + 1
b) ------------8x2 + 4x
x2 + 2x - 8
c) ----------------x2 + 4x
Multiply / Divide:
x2 + x
2x - 2
a) -----------  -----------x2 - 1
4x + 8
x3 + 1
x2 + x
b) ------------  ----------------x2 - x + 1
2x
17
x2 - 2x - 3
c) ---------------x2 - x - 2

x2 - 4x + 3
-----------------3x2 - 3
Add/Subtract
2x – 4
4x – 3
a) --------- + ------------- =
x2 + 5
x2 + 5
b)
3x - 4
----------- 2x + 3
c)
x–4
----------x+ 3
2x - 7
-------2x + 3
-
2x - 1
---------x+3
Other Examples -
x - 1
2
1) ---------- + -------------4x
x
18
2)
2x + 1
---------- 2x2
3)
x + 1
--------x -1
3 + x
4) ---------x +1
5)
2
-------x- 2
3
-------x
x -1
- -----------x+2
x+2
- -------x2 - 1
2 - 2x
+ ---------2 - x
2/x - ½
6) --------------3/x + ½
19
7)
8.
1/(x–1)
--------------------3 - 1/( x – 1 )
1 -
1
--------------------------1
1 - -----------1 - 1/x
20
Solving Equations –
x + 2x = x + 4: solution set ____________
These two equations have the same solution.
x – 2 = 0  solution set _______________
two equations are said to be equivalent provided they have the same solution set.
Reduce equations to an equivalent form whose solution is easy to read – we may use the following two operations
1) You can add (subtract) any quantity to both sides of an equation to obtain an equivalent equation
x + 4 = 3  _____________
x – 5 = - 3  ____________
2) You can multiply (divide) both sides of an equation by any nonzero value
2x = ½  ____________
x/4 = - 2/3  _____________
Other Examples 1) 2x - 4 = 3  _____________
3) 3n/5
-
5) 3/x + ½
7)
2
------- +
x +1
2) x/4
+ 6 = 1  _____________
n = - 2  ____________
4)
= 4/x  _____________
6) ( x – 1 ) / 2 - ( x + 1 ) / 3 = 1  __________
1
-----3
2 - 3( 1 – 2x ) = 1  ____________
1
= ------x+1
21
8 ) another example
22
Quadratic equations –
An equation of the form ax2 + bx + c = 0 is called a quadratic equation in x
ex. x2 = 5x – 2  a = _____ b = _________ c = __________
ex. 1 – x2 = 2x  a = ______ b = _________ c = __________
To find the solution of equations of this form we may use one of the following methods:
1) factoring: either ax2 + bx = 0, or ax2 + bx + c = 0
2) square root method: ax2 + c = 0
3) completing the square: any equation of the form ax2 + bx + c = 0
4) Quadratic formula: any equation of the form ax2 + bx + c = 0
23
Short - Quiz #1
Name ___________________________________________ Math 1302 - – January 17, 2002
1.
Label each of the following sets by using
Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers.
a) { 0, 1, 2, 3, … }  _____________________________________________
b) { a/b : a is an integer and b is a nonzero integer }  _______________________________
c) the set is made up entirely of all rational and all irrational numbers  _______________________
2. Find the smallest natural number. ________________
3. What is the smallest positive integer ? _________________
__
4. What kind of real number is \/ 5 ? A real that is also a _______________________________________
5. Which of these examples illustrates the commutative law of addition ? ___________________
3+4=4+3
( 3 + 4 )+ 5 = 3 + ( 4 + 5 )
3 ( 4 + 5 ) = 3(4) + 3 (5)
6. Which of the following examples illustrates the associative law of multiplication ? ___________________
3(4) = 4(3)
(34)  5 = 3(4  5 )
3(4 + 5 ) = 3(4) + 4(5)
7. Find the absolute value of
a)
| - 4 | = ________________
b. - | - 2 | = _____________
8. How many whole number solutions does the equation
How many rational solutions does the equation
9. Find GCF ( 12, 28 ) = __________________
x2 = 16 have ? ______________________
__
x + \/ 3 = 0 have ? ________________
LCM ( 20, 24 ) = _______________
24
Math 1302 ----------Week 2 – Spring Semester 2002
Other Radicals:
a)
b)
c)
d)
25
Name ___________________________________ Math 1302
Jan. 22, 2002 LongQZ # 1
Note: to grader --- each blank is worth ½ point --- adds up to 16 points – start off everybody with 4 points for a total
of 20 points.
1.
Use the terms that follow to identify each of the sets
set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,
set of real numbers, set of prime numbers
{ 1, 2, 3, 4, 5, ... } is the set of _____________________
Natural numbers that are divisible only by 1 and themselves and are greater than 1 are called ___________________
The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________
Another name for the set of positive integers is the set of _______________________
All real numbers that can not be written as fractions make up the set of _____________________
The real numbers are made up entirely of what two sets? ___________________ and ___________________
2. Identify which law(property) is being used. ( write the complete description; commutative law of addition)
commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law , or None of these
2( x + y ) = 2x + 2y ==> _____________________________
x(y) = y(x) ==> _____________________________
x + (y + z ) = (x + y) + z ==> __________________________
12 – 4 = 4 - 12 ==> ________________________
3. Find each of the following absolute values – exact values – no calculators needed.
-
| 2 +  | = ______________
| - 6 | = _________
4. Find each of the following
a. GCF ( 12, 8 ) = ___________
b. GCF ( 20 , 15 ) = _________
c. LCM ( 8, 6 ) = ___________
5. Simplify each of the following exponent
a.
x  x = _________
4
8
b.
4
(x )
5
= _________
c.
x4
---------- = _________
x12
x12
d. -------- = _________
x8
26
e. ( -2x2y )2 = ___________
6. Simplify. Assume that x  0.
a) - 4 2 = _______________
b) 03 = __________________
c) ( 4x3 )0 = ____________
d) 4 - 2 = ________
7.
Simplify. Leave answers with only nonnegative exponents.
a) (-2x3y - 3 )2 = _____________
b) ( -2xy - 2 )( - 3x2 y2 ) = ___________
c) ( ¾ ) – 1 = ___________
8. More exponents.
1/2
a) x
 2x
3/2
= __________
4x1/2
b) --------------- = ____________
8x3/2
c) ( 41/2 )3/2 = __________
9. One last exponents.
27
Name _________________________________ Math 1302 Quiz, January 24, 2002 ---1.
Find the square roots of 100. __________________
Find the cube root of -27 . _____________
2. Find
__
a) \/ 81 = ___________________
4 ___
b) \/ 16 = _____________
_______
c) \/ 25  49 = ______________
3. Find 4 –2 = ______________
Short Quiz ( 10 points)
____
d) \/ - 4 = ____________
( ¼ )–1
= _________________
4. If x = -2 and y = 0 , then find
x2 - xy = ___________
5. Find
x1/3  2x1/2 = _________________
28
Long Quiz 20 points
Name ____________________________________ Math 1302 – January 29, 2002 -(Note: count each blank 1 point except for #9. Count #9 as three points )
State the five rules of exponents –
1.
a) xm  xn = ___________
d) ( x  y )n = _______________
b) xm  xn = ______________
c) ( x m )n = ___________
e) ( x / y ) n = _______________
2. Simplify using the rules of exponents. All exponents should be positive. Write in simplest form with no radicals.
a) x1/2 x1/3 = ____________
b) ( 4x6 ) ½ = ______________
c) ( - 8x1/3y- 6 )-3 = __________________
d) ( - 8/27 )-1/3 = _____________
3. Write each of the following radicals in simplest form .
3 ________
a) \/ -8x6y12 = ___________________
c)
__________
\/ 900 + 1600 = _______
___
b) \/ 40 = ___________
d) ( - 64x9y
- 12 -1/ 3
)
= ____________
4. More radicals. Write in simplest form.
8 ___
a) \/ x4 = ____________
c)
_____
\/ 8x3y5
= ___________
4 ______
b) \/ 4x2y4 = ___________
__
d) \/ 8 +
__
\/ 50
= __________
29
9.
Name _______________________ Math 1302 – January 31, 2002 --- SHORT Quiz – 10 points
1.
A polynomial with two terms is called a ___________________
2. How many terms does the following polynomial have ? _____________________
3. What is the degree of each of the following polynomials
a) 2 – x  ___________
b) 4 + 2x3 - 6x + y  _________
c) 3  _____________
4. Which of these is not a monomial ? ____________________
2,
- 4x ,
x2y ,
1 - 2x , all are
5. Add the following polynomials.
( 3x2 + 4x - 2 ) + ( 2x2 – 4 ) = _________________________________
6. Find the difference of the following polynomials.
( 2 – x – x2 ) - ( 4 + 5x – 2x2 ) = _________________
7. What is the product of
a) 2 ( x2 - 4x + 4 ) = _________________
b) 2xy2 ( 3 – 2x + xy ) = ________________________
c) ( x + 2y)(4x + 3y ) = ____________________
30
d) ( x + 2y )2 = ____________________________
Name __________________________________ Math 1302 - February 5, 2002 --- LONG
Quiz – 20 points
0. 1 Simplify to a single number.
40 = ____________
1.
-22 = _______________
4 / 0 = ____________
Use the rules of exponents to simplify.
a) 2x3y  4xy4 = ________________
4x-2 y3
c) ( ---------------2x3y -2
( -2x2y4 )3 = ______________________
)2 = ____________________
2. Use the rules of radicals to simplify
3 _____
a) \/ 8x6y12 = __________________
4 ____
b) \/ 4x2 = ____________
3. More radicals
4
a) -------- =
\/ 2
c)
-----------------
4. What is the degree of the following polynomials.
a) 210 - x  _______________
b) x4 + y3x2  _____________
5. perform the given operation and reduce to lowest terms (simplest ) .
a) 4x(x2 - 2x ) = ____________________
b) ( x – 2y)(x + 2y ) = ____________________
6. Find GCF( 12, 40 ) = _______________
GCF ( 4x2y, 10xy3 ) = ______________
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7. Factor each of the following polynomials.
4x – x = _________________
8xy – 2y2 = _________________
2
2
__________
x - y + 2y - 1 = ___________________________
Name ________________________________ Math 1302 - February 7, 2002 ---- Short Quiz ( 10 points )
Factor each of the following polynomials.
1) 2x + x = ___________________
2) x ( y – x ) - y ( y – x ) = _____________________
3) x3 - 4x = ______________________________
4) x3 + 64 = _____________________________________
5) x4 - 16 = ________________________________________
6) x2 + 6x + 5 = ___________________________________
7) x2 + 12x + 36 = __________________________________
32
8) x2 + 4 = _________________________________________
More examples:
1) 8/89.
Sally took four tests in science class . On each successive test, her score improved by 3 points. If her mean score
was 69.5 %, what did she score on the first test ?
2) 12/89 A college student earns $20 per day delivering advertising brochures door-to-door, plus 75 cents for each person he
interviews. How many people did he interview on a day when he earned $56.
3) 16/89 A builder wants to install a triangular window with the angles shown below. What angles will he have to cut to
make the window fit ?
Other example:
x -1
-----2
-
x + 1
---------3
= 1
Test I review:
 Know sets of numbers – natural #’s, whole #’s, integers, rational, irrational, real, prime, nonnegative integers, positive
integers
 be able to classify a number as a member of a particular set
 be able to give an example of a number that is a member of one set but not another
 compare sets, know how they differ in terms of the elements (numbers) in the set
 Properties of sets –
33
 commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law
 Properties and definitions of numbers
absolute value , GCF, LCM
inequalities on a number line
order of operations: PEMDAS
scientific notation
evaluating expressions
 basic operations
be able to add, subtract, multiply, divide any real number
 exponents
know the five basic properties
work with positive integer exponents , exponent of zero, negative coefficients or signs , negative exponents, fractional
exponent
 radicals –
be able to simplify and work problems as done in class and over homework ,
fractional exponents and radicals
 polynomials –
basic operations - add, subtract, multiply, divide,
degree, terms, factors, classify – monomials, binomials, trinomials,
long-hand division
 factor polynomials; GCF, difference of squares, perfect squares, sum of cubes, difference of cubes, trinomials, grouping
make sure to work with any combination of the above methods
 algebraic fractions: add, subtract, multiply, divide,
complex fractions
 linear equations in one variable - solve, find solution set
solve for a variable in a formula
word problems
 Identify equations as linear, quadratic equations
state methods of solutions for quadratic equations
Write a quadratic equation in its standard form, identify a, b, c
Write down quadratic formula
Name _______________________________ Math 1302 – February 12, 2002
1.
Long Quiz 20points
Factor each of the following polynomials.
a) x2 – 16y4 = ________________________
34
b) 2x2 - 5x + 4 = ___________________
c) 8x4 - xy3 = ________________________
2. More factoring
a) 4( x – 2y ) + 2y ( x – 2y ) = _________________________
a) 2x – xy + 6y - 3y2 = ___________________________
3. Reduce each of the following fractions .
2x
a) ------------ =
2x2 - x
b)
x3 + 8
----------------- =
x2 - 2x + 4
HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,
Name ___________________________________ Math 1302.F10
1.
Jan. 17, 2001 Quiz # 1
Use the terms that follow to identify each of the sets
35
set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,
set of real numbers, set of prime numbers
{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________
Natural numbers that are divisible only by 1 and themselves and are greater than 1 are called ___________________
The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________
Another name for the set of positive integers is the set of _______________________
All real numbers that can not be written as fractions make up the set of _____________________
The set of _______________ is made up of only rational and irrational numbers.
2. Identify which law(property) is being used.
commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law , or None of these
2( x + y ) = 2x + 2y ==> _____________________________
x(y) = y(x) ==> _____________________________
x + (y + z ) = (x + y) + z ==> __________________________
12 – 4 = 4 - 8 ==> ________________________
3. Find each of the following absolute values
| 9 | = ____________
-
| - 2/ 3 | = ______________
| 2 +  | = ______________
| - 6 | = _________
4. Find each of the following
a. GCF ( 12, 8 ) = ___________
b. GCF ( 20 , 15 ) = ______________
c. LCM ( 8, 6 ) = ___________
5. Simplify each of the following exponent
a.
x  x = _________
4
8
b.
4
(x )
5
= _________
c.
x4
---------- = _________
x12
Name ___________________________________ Math 1302.F10
x12
d. -------- = _________
x8
Jan. 17, 2001 Quiz # 2
HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,
1.
Use the terms that follow to identify each of the sets
set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,
36
set of real numbers, set of prime numbers
{ 1, 2, 3, 4, 5, ... } is the set of _____________________
The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________
All real numbers that can be written as fractions make up the set of _____________________
2. Identify which law(property) is being used.
commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law , or None of these
2( x + y ) = 2x + 2y ==> _____________________________
x + (y + z ) = (x + y) + z ==> __________________________
3. Find each of the following absolute values
a. | 2 -  | = ____________
b. - | - 2/3 | = ___________
4. Find each of the following
a. GCF ( 12, 24 ) = ___________
b. LCM ( 12 , 24 ) = ______________
5. Simplify each of the following exponent
4
8
a. ( - 2x ) (3 x y) = ____________
b.
4
( 2x )
5
= _________
c.
8x4
---------- = _______
6x12
6. Find
a.
b.
e.
f.
c.
d.
g.
W: p. 12: 1 – 20 all (except for 15-17), 21, 26, 31, 36, 41, 46 51, 56, 61, 66, 67, 71,
p 24:
1-12 all, 15, 33, 36, 39, 42, 45, 48, 51,
HW: p. 24: 54, 57, 60, 63, 66, 69, 72, 82, 91, 94, 97, 119, 122, page 39: 11, 14, 17, 20,
Name ___________________________________ Math 1302.F10
1.
May 31, 2001 Quiz # 2
Use the terms that follow to identify each of the sets
37
set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,
set of real numbers, set of prime numbers
{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________
The set that contains the whole numbers and their opposites is called is called the set of ___________________
All real numbers that can not be written as fractions make up the set of _____________________
2. Identify which law(property) is being used.
commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law , or None of these
x  (y  z ) = (x  y)  z ==> __________________________
xy = yx _____________________________
3. Find each of the following absolute values
a. | 4 -  | = ____________
b.
| x | = ___________ if x is a negative integer
4. Find each of the following
GCF ( 30, 24 ) = ___________
5. Simplify each of the following exponent
4
8
a. ( - 2x ) (3 x y) = ____________
b.
4
( 2x )
5
= _________
8x4
---------- = _______
6x12
c.
6. Find
a. – 4
e.
-2
= _________
b. – 16 –1/2 = _________
–8
c.
2/3
= ________
( - 16 ) ½ = ________
7. Simplify.
a. ( 2x3y-2)-3 = __________
b. \/ 64 = _________
c.
2 \/ 25 - 4 \/ 36
Name ___________________________________ Math 1302.F10
1.
Jan. 17, 2001 Quiz # 4
Use the terms that follow to identify each of the sets
{ 1, 2, 3, 4, 5, ... } is the set of _____________________
38
The set that contains the whole numbers and their opposites is called is called the set of ___________________
All positive integers _____________________
2. Identify which law(property) is being used.
x  (y  z ) = (x  y)  z ==> __________________________
x+y = y+x _____________________________
3. Find each of the following absolute values
a. | - 4 -  | = ____________
b.
| x | = ___________ if x is a whole number
4. Find each of the following
LCM ( 30, 24 ) = ___________
5. Simplify each of the following exponent
-4
-8
-2
= _________
a. ( - 2x ) (3 x y) = ____________
b.
44
( 2x )
-5
= _________
8x4y-2
c. ---------- = _______
6x-12 y-4
6. Find
a. ( – 4 )
b. ( – 16 –1/2) = _________
c.
–8
-2/3
= ________
7. Simplify.
a.
b.
c.
8. what is the degree of the following polynomials ?
a.
b.
c.
Math 1302 – Quiz # 3
1. Use the numbers; 0, - 2, 3, ½, 0.01001000100001... to answer the questions that follow.
Give me an example of
a) an integer that is not a whole number ______________
b) a rational number that is not an integer _______
39
2. Which property is being use;
( a + b) = ( b + a ) _______________________________________________
3. Use the rules of exponents to simplify.
a) - 4 2 = _____________
d)
- 40 = ____________
g) - 163/4 = ___________
= ______________
c) 0 4 = _______________
e) - 251/2 = ______________
f) 0 / 4 = ______________
b) 16
-1 / 2
h) ( - 8/ 27 ) 2/3 = _______________
4. Use the rules of radicals to simplify.
a)
16x8y12 = ______________
c) \/ 8x3y7 = ______________
e) 3 \/ 16
g)
- 4 \/ 4 = _______
\/ 8x3
b)
d) \/
f)
= _____________
4
--------9x
3 \/ 50 = _________
4
---------\/ 8
5. Use the rules of exponents to simplify.
a) ( - 8x6 y – 9 )
b)
- 1/3
= ______________
16x – 1 y2
( ------------------- ) 2
12 x -2 y – 2
c)
40
Rational Exponents
More on Radicals
HW
page 15: 1 – 15 all, 25, 28, 33, 47, page 20: 1 – 16, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 46a, 46b,
Name ___________________________________ Math 1311.030
1.
Jan. 22, 2001 Quiz # 1
Use the terms that follow to identify each of the sets
set of natural numbers, set of whole numbers, set of integers, set of rational numbers, set of irrational numbers,
set of real numbers, set of prime numbers
{ 0, 1, 2, 3, 4, 5, ... } is the set of _____________________
Natural numbers that are divisible only by 1 and themselves and are greater than 1 are called ___________________
The set { ........... – 3, - 2, - 1, 0, 1, 2, 3, ... } is called the set of ___________________
Another name for the set of positive integers is the set of _______________________
All real numbers that can not be written as fractions make up the set of _____________________
The set of _______________ is made up of only rational and irrational numbers.
2. Identify which law(property) is being used.
commutative law of addition, commutative law of multiplication, associative law of addition, associative law of
multiplication , distributive law , or None of these
2( x + y ) = 2x + 2y ==> _____________________________
x(y) = y(x) ==> _____________________________
x + (y + z ) = (x + y) + z ==> __________________________
41
12 – 4 = 4 - 8 ==> ________________________
3. Find each of the following absolute values
| 9 | = ____________
-
| - 2/ 3 | = ______________
| 2 +  | = ______________
| - 6 | = _________
4. Find each of the following
a. GCF ( 12, 8 ) = ___________
b. GCF ( 20 , 15 ) = ______________
c. LCM ( 8, 6 ) = ___________
5. Simplify each of the following exponent
a.
x  x = _________
4
8
b.
4
(x )
5
= _________
c.
x4
---------- = _________
x12
x12
d. -------- = _________
x8
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