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Prerequisite Chapter Homework Assignment Sheet
Section P1: Review of Real Numbers and Their Properties
Assignment 1: Pg 9 – 11 #’s 2 – 60 evens, 68 – 84 evens, 98 – 104 evens
Section P2: Exponents and Radicals
Assignment 2: Pg 21 – 22 #’s 2 – 36 even #’s
Assignment 3: Pg 22 #’s 51 – 62 all
Assignment 4: Pg 22 #’s 64 – 78 even #’s
Assignment 5: Pg 22 – 23 #’s 80 – 102 even #’s
Assignment 6: Pg 21 – 23 #’s 5 – 11, 25 – 35, 69 – 73, 91 – 101 ONLY ODD #’S
Section P3: Polynomials and Special Products
Assignment 7: Pg 29 - 30 #’s 2 – 6, 12 – 46, 56 - 60 Only even #’s
Assignment 8: Pg 30 – 31 #’s 62 – 96 even #’s Plus 103 a and 105
Section P4: Factoring Polynomials
Assignment 9: Pg 38 #’s 2 – 46 even #’s
Assignment 10: Pg 38 #’s 51 – 64 all
Assignment 11: Pg 38 – 39 #’s 65 – 78 all
Assignment 12: Pg 39 #’s 79 – 111 odd #’s
Section P5: Rational Expressions
Assignment 13: Pg 48 #’s 2 – 28 even #’s
Assignment 14: Pg 48 #’s 35 – 42 all
Assignment 15: Pg 49 #’s 44 – 60 even #’s
Assignment 16: Pg 48 – 49 #’s 7, 21, 25, 27, 43, 49, 51, 55, 57, 61 – 64 all, 67
Section P6: Errors and the Algebra of Calculus
Assignment 17: Pg 56 #’s 1 – 18 all
Assignment 18: Pg 56 – 57 #’s 19 – 22 all, 43 – 48 all, E.C. 60
Section P7: The Rectangular Coordinate System and Graphs
Assignment 19: Pg 64 #’s 2 – 20 even #’s, 28, 30, 32 – 40 only b and c, 41, 48, 52
Review Prerequisite Chapter
Assignment 20: Pg 70 – 71 #’s 2 – 66 evens except 18, 20, 32, 34
Assignment 21: Pg 71 – 72 #’s 67 – 86 all, 90, 92, 96, 99, 106, 108, 110, 112
2
P1: Review of real numbers and their properties
A.
Real numbers
B.
Ordering real numbers
Ex.
Put the following in order on the number line.
5 6
2
1
5,  4,  9.2,  9.05,  9.1, , ,.50, 4 , 4
9 13
5
7
Ex.
Put the following on the number line.
a.
x 3
b.
7  x  6
3
There is another way of interpreting inequalities, describing them as __________ of real
numbers called intervals. In bounded intervals like below, the real numbers a and b are the
______________ of each interval.
Bounded intervals on the real number line
Notation
 a, b 
Interval Type
inequality
a xb
 a, b 
a xb
[a, b)
a xb
(a, b]
a xb
Graph
Unbounded intervals on the real number line
Notation
[ a,  )
Interval Type
inequality
xa
 a,  
xa
( , b]
xb
 , b 
xb
 ,  
  x  
Ex.
Graph
Give a verbal description of the following, write it like an inequality and graph it
a.
 4,9 
b.
[3,8)
c.
(, 4]
4
C.
Absolute Value and Distance
The absolute value of a real number is its magnitude, or the distance between the origin
and the point representing the real number on the real number line.
a, if a  0
Def: | a | 
a, if a  0
Properties of Absolute Values
1.
| a | 0
2.
| a || a |
3.
| ab || a || b |
Ex.
1.
| 3 |  | 9 |
3.
|12  19  8 |
4.
a
a
 ,b  0
b
b
2.
4.
| 10  9 |
| 7  29 || 29  7 |
5
P2. Exponents and Radicals
Day 1
A.
Integer Exponents
Let’s complete the following table that shows the properties of exponents.
Properties
1. x m  x n 
Solution
Ex
244  245 
2. ( x m )n 
p 
3. ( xy)m 
3x 
4. x  m 
62 
5. x 0 
100, 000,203, 0450 
xm
6. n 
x
48

45
 x
7.  
 y
2 5

4

5
u  
 
v 
m
8. | a 2 |
|  3 |
2
**Simplified means
no
no
no
no
( )
common bases
decimals
negative exponents unless told otherwise
Examples:
Evaluate each of the following using the rules of exponents.
1. (4)(4)3
2. [(3)2 ]3
3. (32 x 2 y 8 )2
4. 54  53
5. (23 )2
6. m 7 
1
m4
6
7.
3
83  85
=
89
5
8.   =
6
9.
5x 4 3x 3 y5
=

8 x5 6 y 4
2
10.
3 4 
4
3
11.
4 4 
2
3
12.
 9 x 2 y 6 
 3 6  
 4x y 
Day 2
B.
Radicals and their properties
Principal nth Root of a Number:
Let a be a real number that has at least one nth root. The principal nth root of
a is the nth root that has the same sign as a . It is denoted by a radical symbol
n
a
The positive integer n is the index of the radical, and the number a is the
radicand. If n = 2, omit the index and write a rather than 2 a . ( The plural of
index is indices).
To simplify the following radicals with different indexes, you need to keep in mind such
things as not only perfect squares, but also perfect cubes, etc of both positive and
negative integers.
7
Ex.
Simplify the following without using a calculator.
1.
49
2.
4.
3
27
5.
7.
3
64
8.
 25
4
25
16
3.
16
6.
5
243
64
9.
4
81
3.
4
16 x 4 
So, if a  0 and n is even, ____________.
if a  0 and n is odd, _____________.
Properties of Radicals
Property
1.
n
a 
2.
n
an b 
3.
Ex
3
3
a

n
b
m n
5.
 a
n
162

6
3
4 2
a 
n
27 4 
32  2 
n
4.
6.
Solution
m
7

n even:
n
xn 
n odd:
n
xn 
17 
127 
x2 
5
x5 
Examples:
1.
2

18
4.
 
5
12
5

2.
3
27 x 3 
5.
3
93 3
8
Day 3
C.
Simplifying Radicals
Radical in the simplest form must satisfy the following conditions:
1.
All possible factors have been removed from the radical.
2.
All fractions have radical-free denominators. (must rationalize the denominator)
3.
The index of the radical is reduced.
1.
D.
20 
2.
 50 
3.
4.
3
64x5 
5.
3
32x6 
6.
7.
5
96x 5 y12 
8.
4
32
9.
3
250 
8x4 
3
32x6
Combining Radicals
Radicals can be combined only if they have the same index and radicand. You may have to
simply the radical to eventually combine them.
Ex.
1.
4 3 6  12 3 6  5 4 6
2. 5 3 2  3 16 
8.
3
54  3 81 
9.
4
32  2 4 16  3 4 162 
9
Day 4
E.
Rationalizing Denominators and Numerators
When simplifying a rational expression, you cannot leave a radical in the denominator. You
must eliminate the radical by multiplying the denominator by the conjugate of the form
of a  b m or a  b m .
Ex.
4
6
1.

2.

3
3 10
4
8
2 3
3.
F.
Rational Exponents
1
1
is the rational exponent of a.
a n  n a , Where
n
m
 1n 
Additionally, a   a  
 
m
n
Ex.
 a
n
m
m
n
1
3
5 
2.
1
and a   a m  n  n a m
Rewrite in Radical notation
1.
Ex.
12
5 2
4.
Ex.
Rewrite in exponent notation
1
6
12 
5
3.
19 
4.
9
28 
Simplify the following rational exponents
1
1
1. 53  5 4 
2.
1
4
3.
 8
3
2

15
3

 15 13 
4.  2  2  


10
5.
3
6.
125
x 3
 x  3

2
3
P3: Polynomials and Special Products
A.
Polynomials
To effectively manipulate polynomial expressions in solving real-life problems we must be
able to add, subtract, multiply and divide the terms that make-up the polynomials and put your
polynomial in standard form, which is writing the polynomial in descending order.
Let’s classify the next 5 polynomials and identify the: _______________, ______________,
______________, ________________ and _________________.
Polynomial
5
3x 2
4x  1
x  2  5x 3
8x 4  7x 2  11x 9  4x  9
Note: Notice polynomials involve only one variable with non-negative integer exponents.
Is the following a polynomial: x 3  8x 2 
4
 9 ? ___________________
x
When adding and subtracting polynomials remember to only combine ____________ terms.
Ex 1:
A.
(5x 2  4x  8)  (3x 2  2x  3)
B.
(8x 2  14x 3  4x  2)  (6x 3  4x 2  7)
11
Ex 2: Multiplying by a monomial
C.
4x 2 (6x 3  7x  5)
Ex 3: Multiplying Polynomials (Horizontal v’s Vertical)
D.
(x  5)(2x 3  3x 2  4x  1)
Horizontal
B.
Vertical
Special Products
Some binomial products occur so frequently that it is worth memorizing their special
product patterns:
Sum and Difference of same terms

(u  v )(u  v )  u 2  v 2
Square of a binomial

(u  v )2  u 2  2uv  v 2

(u  v )2  u 2  2uv  v 2
Cube of a binomial

(u  v )3  u 3  3u 2v  3uv 2  v 3

(u  v )3  u 3  3u 2v  3uv 2  v 3
Ex 4: Recognizing Patterns
A.
(7 x  4)(7 x  4) 
B.
(5  6x )2 
C.
(2x  9)3 
D.
(4x  7)3 
12
C.
Application
Write a polynomial that expresses the area of the shaded region below
2x-1
x
3x
4x+3
P4: Factoring Polynomials
A.
Polynomials with common Factors
When it comes to removing a common factor, we need to think back to Distributive
property: a(b  c)  __________ . Now, let’s look at the Distributive property in reverse
direction: ab  ac  ___________ . Notice the connection? Lets look at the following
examples.
Ex.
Factor each expression by finding its common factor.
1.
8 y4  6 y
2.
9 x 3  54 x  27
3.
B.
8 x 4  26 x 2  14 x
4.
 x  3 x   x  3 9
Factoring Special Polynomials Forms
Remember the special product forms we learned in P3, well here is the reverse form when
it comes to factoring.
13
Difference of Two Squares: u 2 v 2  (u v )(u v )
1.
x 2  52 
2.
3x 2  48 
3.
x 2  36y 2 
4.
5  45x 2 
5.
x
6.
16  81x 4 
2
 5   16y 2 
Perfect-square factoring: a.
u 2  2uv  v 2
b.
u 2  2uv  v 2
1.
x 2  4x  4 
2.
x 2  6x  9 
3.
x 2  10x  25 
4.
x 2  8x  16 
5.
16x 2  24x  9 
6.
9x 2  42x  49 
14.
3x 3  192 
Sum of Two Cubes: u 3 v 3  (u v )(u 2  uv v 2 )
13.
x 3  27 
15.
64x 3  27 y 3 
Differences of 2 cubes: u 3 v 3  (u v )(u 2  uv v 2 )
16.
x 3  27 
18.
8x 3  216 
17.
125x 3  8 
14
C.
D.
Trinomials with Binomial Factors
There are many ways to factor trinomials. No matter what method you use, they are all
guess and check. I personally use the box method but you can use any method you like.
Ex.
Factor the following trinomial when L.C. is 1 & using master product sum.
1.
x 3  5x 2  6x
Ex.
Factor the following polynomials when L.C. is not 1.
4.
2x 2  7x  3
2.
5.
2x 2  16x  32
2x 2  x  3
3.
6.
x 2y  2xy  1y
3x 2  11x  20
Factoring by Grouping
Ex.
Factor the following polynomial using grouping method.
7.
2x 3  3x 2  4x  6
Ex.
10.
Factoring a trinomial using the grouping method.
2 x2  5x  3
11.
4 x 2  12 x  9
8.
5xy  10x  15y  30
9.
xy  2x  3y  6
15
P5: Rational Expressions
A.
Domain of an Algebraic Expression
By finding the domain, you are looking for the set of x-values that defines the
expression.
Ex.
Which of the following functions have restrictions on their domains?
a.
f ( x)  4 x  9
Restriction:
Restriction:
c.
h( x )  2 x  6
l ( x)  6 x  18
d.
Restriction:
Restriction:
e.
m(t ) 
x3
x4
f.
Restriction:
n( x ) 
5
4 x  32
Restriction:
g.
B.
g ( x)  4 x 2  7 x  1
b.
r ( x) 
3x  4
9
Restriction:
Simplifying Rational Expressions
Reducing Rational Expressions: Factor the numerator and denominator first and then
divide out common factors. Also denote what values x
cannot equal to.
1.
8x 2y

24xy 5
2.
8x 3  2x 2

4x 2  x
3.
x 2  49

x 2  8x  7
16
C.
1.
4.
Operations with Rational Expressions
Multiplying Rational Expressions: Factor first and then divide out common factors.
8x 3 14y 4
2x 2  x  3 x 2  x
x 2  7x  8 4x 3


2.


3.


26xy 12x 2
x2 x
2x  3
3x 2  24x x 2  1
Dividing Rational Expressions: Change the division to multiplication of the reciprocal.
3x  1
x
x 2  4x  4
x 1


5.
  x  2 
6.
(x 2  1) 

3
3
2
2x  2x 2x  2x
x 3
x 5
Adding and subtracting rational expressions
-
A.
Find the least common denominator (LCD)
Rewrite each rational expression using the LCD
Add or subtract the numerator
Reduce answer when possible
5
2
 2 
x 3x
B.
3
4


x  5 x 1
17
C.
D.
A.
C.
x
3x
 2

x 1 x 1
D.
5
3x  1
2
 2
 
3x  12 x  x  12 3
Complex Fractions and the Difference Quotient
Simplifying Complex Fractions – Multiplying numerator and denominator by the LCD of
every fraction.
1

2 
x


x
 6

 3

 x 1  
3
x
B.
 1
1  
 x 
3 

1  2 
 2x 
D.
 x2 
 2 
 x 1  
3x
x 1
18
E.
Simplifying expressions with negative exponents
1.
5

3
3x  2  5 x   (2  5 x)

2
3
2.
 x3 1  x 2 

1
2
1
 2 x 1  x 2  2
x4
P6: Errors and the Algebra of Calculus
A.
1.
Writing a fraction as a sum of terms
x 4  9 x3  8x 2  12 x  18
2.
3x
8x  9 x 2  12
x
19
P7: The Rectangular Coordinate System and Graphs
A.
The Cartesian plane.
Graphs of linear equations are geometric representations of the relationship between the
variables. In this lesson we will use the coordinate plane to visualize this relationship
between the variables.
Ex. Coordinate Plane
On the coordinate plane below, label each axis, each quadrant and origin.
10
Plot the following ordered pairs
a)
(3, 0), (-2, 0), (-10, 0), (8, 0)
What kind of line is formed?
b)
What kind of line is formed?
-10
-10
B.
(0,8), (0,3), (0,-2), (0, - 10)
10
The Distance Formula: It is derived from the Pythagorean Theorem.
( x1 , y1 ) and ( x2 , y2 ) is d 
.
20
C.
Ex.
Find the distance between (-2, 12) and (5, 7).
Ex.
Verify that the following three ordered pairs creates a right Triangle.
 2,1 ,  4,0 , 5,7 
The Midpoint Formula:
It is used to find the Mid-point between two points
 x1, y1  &  x2 , y2  .
 x  x y  y2 
Midpoint   1 2 , 1

2 
 2
Ex.
Find the midpoint between the following segment:  4, 8 &  1,10
21