Download Transition to College Math Review Notes Name R.1 Algebra and

Transition to College Math
Review Notes
Name ___________________________________________________________________
R.1 Algebra and Real Numbers
Essential Question(s):
 How do you graph real numbers on a number line and add, subtract, multiply, and
divide real numbers?
Definitions:
Set
A collection of objects. Note: Two sets are equal if they have exactly the same
objects)
Element (member)
Each object in a set
Finite
Elements can be counted
Infinite
Elements can be counted without end
Empty (null) Set
 
Set that contains no elements { }. Note: the empty set is a subset of any set.
Listing (roster) Method
Lists the elements of a set. Ex: A = { 1, 2, 3, 4 }
Set Builder (rule) notation
Uses a rule to describe the members of the set. Ex: {x | x > 5} (Read “x such that
x is greater than 5”)
Variable
A symbol used as a placeholder for the elements of a set with two or more
elements
Replacement Set
The elements of a set represented by a variable
Constant
A symbol that names exactly one object
Subset
Element(s) that are members of a set

means…
“is an element of…”
Page 1
Transition to College Math

means…
Review Notes
“is a subset of…”
Equal Sets
Two or more sets having exactly the same elements
Intersection of Sets A  B
The set of elements common to both sets A and B
Union of Sets A  B
The set of elements that are members of set A or of set B or of both sets
natural numbers
{1, 2,3,...}
whole numbers
{0,1, 2,3,...}
integers
{..., 3, 2, 1, 0,1, 2,3,...}  the whole numbers and their opposites
rational numbers
irrational numbers
real numbers
numbers than can be written as a quotient of two integers → rational means
fractional (decimals terminate or repeat)
5 7
3 9
Ex.  , , 0.5,
4
1
numbers that are not rational (nonrepeating and nonterminating decimal
numbers) Ex.
7,  ,  13
rational numbers and irrational numbers
Examples:
1. List all possible subsets of A = {1, 2, 3}
{1, 2,
3}
{1, 2}
{1, 3}
{2, 3}
{1}
{2}
{3}
0
2. Determine if the following are true or false
a. 2  {1, 2,3}
F
d. 0  {x, y, z}
F
b. { y, x, z}  {x, y, z}
T
e. {w, x, y, z}  {x, y, z}
F
c. { y, x, z}  {x, y, z}
T
Page 2
Transition to College Math
Review Notes
3. Write the following sets using set builder notation
a. {2, 4, 6, 8, 10}
{x | x an even integer between
2 and 10 inclusive}
b. {1, 2, 3, 4, 5}
{x | x an integer between 1
and 5 inclusive}
4. Write the following set using the listing method:
{x | x is a letter in the word ALGEBRA}


5. Given set B   2,
1
,
3
2, 0,
8
, ,
3
a. Rational numbers
{A, L, G, E, B, R}

36  , list the following

1
8

2, , 0, ,
3
3


b. Irrational numbers
2, 


36 

2, 0, 36
0, 36
 36
c. Integers
d. Whole numbers
e. Natural numbers
6. Find the intersection: {1, 2, 3, 4, 5, 6}  {2, 4, 6, 8}
7. Find the union: {1, 2, 3, 4, 5, 6}  {2, 4, 6, 8}
{2, 4, 6}
{1, 2, 3, 4, 5, 6, 8}
Define or describe the following:
Graph of a number
The point on a number line that is paired with the number
Coordinate of a point
The number paired with a point on a number line
Origin
The zero point on a number line
Page 3
Transition to College Math
Review Notes
Real Number
Any number that is positive, negative or zero.
Positive Real Number
A number paired with a point on the positive side of a number line
Negative Real Number
A number paired with a point on the negative side of a number line
> means … greater than
Inequality Symbols:
Examples.
< means … less than
4
1. Graph the numbers 1.5,  , 2, 4 , 3, and 2.75 on a number line. Then list them in
3
increasing order.


-4
-3

-2
 
-1
0
1
2

3
4
4,  2.75,  , 1.5, 2, 3
3
2. State two inequalities, one with > and one with <, for the coordinates of the points
shown below.
-3
-2
-1
0
1
2
3
2  1 OR 1  2
Properties of Equality
Statement
For all real numbers a, b, and c.
Reflexive Property
aa
If a  b, then b  a.
Symmetric Property
If a  b and b  c, then a  c.
Transitive Property
If a  b, then a  c  b  c and c  a  c  b.
Addition Property
Multiplication Property
If a  b, then ac  bc and ca  cb.
Field Properties of Real Numbers
For all real numbers a, b, and c.
Closure Properties:
a + b and ab are unique real numbers
Examples
Let a = 12, b = 6, and c = 5.
12  6  18
12(6)  72
Page 4
Transition to College Math
Review Notes
12  6  6  12
Commutative Properties:
a+b=b+a
ab = ba
Associative Properties:
(a + b) + c = a + (b + c)
(ab)c=a(bc)
Identity Properties:
There are unique real numbers 0 and 1
such that:
of Addition: a + 0 = a and 0 + a = a
of Multiplication: a 1  a and 1 a  a
Inverse Properties:
Property of Opposites: For each a, there
is a unique real number –a such that:
a + (-a) = 0 and (-a) + a = 0
12(6)  6(12)
(12  6)  5  12  (6  5)
(12  6)5  12(6  5)
12  0  12
0  12  12
12(1)  12
1(12)  12
12  (12)  0
12  12  0
Property of Reciprocals: For each a
except 0, there is a unique real number
1
a
12 
such that:
1
1
a   1 and  a  1
a
a
Distributive Property:
a(b  c)  ab  ac
(b  c)a  ba  ca
1
1
12
1
12  1
12
12(6  5)  12(6)  12(5)
(6  5)12  6(12)  5(12)
Zero Properties
a 0  0 a  0  Any number multiplied by zero is zero
ab = 0 iff a = 0 or b = 0  The product ab can only equal zero if a or b or both equal zero.
Properties of Fractions Note: Denom  0
Properties of Negatives Note: b  0
– (– a) = a
(– a)b = – ab
(– a)( – b) = ab
– 1(a) = – a
a c

b d
ka a

kb b
a c a
 
b d b
a
a a
 
b b
Examples: b
a
a
a a



b
b
b b
a c ac
 
b b
b
Page 5
d
c
iff ad = bc
a c ac

b d bd
a c ac
 
b b
b
a c ad  cb
 
b d
bd
Transition to College Math
Review Notes
Examples:
1. Is the set of odd numbers closed under addition? Use an example to justify your response.
Odd + Odd = Odd?
7 + 3 = 10
NO
2. Is the set of odd numbers closed under multiplication? Use an example to justify your response.
(Odd) (Odd) = Odd
(7) (3) = 21
YES
3. Is the set of integers closed under subtraction? Use an example to justify your response.
Integer – Integer = Integer?
–5–3=–8
YES
4. Is the set of whole numbers closed under subtraction? Use and example to justify your response.
Whole number – Whole number = Whole number?
6–9=–3
NO
5. Justify each step using a property in this section
1.
2.
(7  3)  [(7)  (3)]
 [(7  3)  (7)]  (3)
 [7  (7)  3]  (3)
3.
 [0  3]  (3)
4.
 3  (3)
5.
0
Page 6
Associative Property
Of Addition
Associative and
Commutative Properties
Inverse Property of
Addition (Property of
Opposites)
Identity Property
of Addition
Inverse Property of
Addition (Property of
Opposites)
Transition to College Math
Review Notes
R.2 Exponents
Essential Question(s):
 How do you simplify exponents and express numbers using scientific notation?
Integer Exponents
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
3. For n a negative integer:
1
an  n
a
Examples:
1. 6360  1
2. ( x 2 )0  x0  1
3. 105 
1
1

 0.00001
5
10 10000
4.
1
 103  1000
103
5.
1
 x4
x 4
u 7 v3
6. 3  7
v
u
Page 7
Transition to College Math
Review Notes
Properties of Integer Exponents
m
1. a a  a
m
2.  a
n

n m
mn
 a mn
am
a
4.    m
b
b
b0
a mn
am 
5. n   1
a
 nm
a
a0
3. (ab)m  a mbm
Examples:
1. (5 x 3 )(3x 4 )  15 x
9 y 7
3
2.
 3
4
6y
2y
3. 2 x 4  (2 x)4  2 x 4  16 x 4  14 x 4
4. (3x 4 y 3 )2  32 x 8 y 6 
y6
9 x8
3
 x2 
y12
5.  4   6
x
y 
6m2 n3
2n 5
6.

15m1n2 5m
3
 x 3 
7.  4 4   x 9
y y 
8.
1
 ( a  b) 2
2
( a  b)
Page 8
Transition to College Math
Review Notes
Definition of an nth Root
a is an nth root of b if
Example: 2 is a 4th root of 16 since
an  b
2 4  16
Number of Real nth Roots of a Real Number b
n Even
n Odd
b Positive
2
Example:
2 are 4th roots of 16
1
Example:
3 is the cube root of 27
b Negative
0
Example:
– 4 has not real
square roots
1
Example:
– 3 is the only real
root of – 27
1
n
b , Principal nth Root
1
For n a natural number and b a real number, b n is the principal nth root of b, defined as follows:
1
1. If n is even and b is positive, b n then represents the positive nth root of b.
1
2. If n is even and b is negative, then b n does not represent a real number.
1
3. If n is odd, then b n represents the real nth root of b.
1
4. 0 n  0
Page 9
Transition to College Math
Review Notes
Anatomy of a Rational Exponent
5
Base
2
3
Exponent
Index
Examples:
1
1. 4 2  4  2
1
2
2. 4   4  2
1
3. (4) 2  4  not real
1
3
4. 8  3 8  2
1
3
5. (8)  3 8  2
1
8
6. 0  8 0  0
3
2
7. 9  93  9 9  9(3)  27
4
3
8. (27)  3 (27)4  27 3 27  27(3)  81
9 4
13

 3  1 
9.  5 y 4   2 y 3   10 y 12 12  10 y 12  1012 y13  10 y12 y



4
 3 1 
16 y
10.  2 x 4 y 4   16 x 3 y  3
x


 12
8x
11.  2

 x3

3
translates to
1
1
3
6 8
1
6


2
x
2
 
36 36
18

2
x

2
x
 1
2

18

x
x9

Page 10
52
Transition to College Math
Review Notes
Scientific Notation
a 10n
1  a  10, n an integer, a in decimal form
To convert from scientific notation to decimal form:
5.76  106
5.76  106
Exponent negative
Decimal moved to the left
Exponent positive
Decimal moved to the right
0.00000576
5, 760, 000
To convert from decimal form to scientific notation:
0.00000576
5, 760, 000
Original number < 1
Power is negative
Original number > 10
Power is positive
5.76  106
5.76  106
Examples:
Write each number in scientific notation:
1. 430
4.3 102
2. 23,000
2.3 104
3. 345,000,000
3.45 108
4. 0.3
5. 0.0031
3.1103
6. 0.000000683
6.83 107
Write in standard decimal form:
1. 4  103
4,000
2. 5.3  105
530,000
3. 2.53 102
0.0253
4. 7.42 106
0.00000742
3  101
Page 11
Transition to College Math
Review Notes
R.3 Radicals
Essential Question(s):
 What are the properties of radicals?
 How do you simplify radicals and convert to rational exponents?
Anatomy of a Radical
When in Simplified Form:
1. powers of factors in radicand are less
than index
2. power of radicand and index cannot
have common factor other than 1.
3. No radicals in denominator
4. No fractions within radical
Exponents
Index
5
32x 23 y 6
Radicand
Define or describe the following:
n
Principle nth root
of a real number
n
a  b means b  a
If the index is even, than a and b are nonnegative.
If the index is odd, a and b can be any real numbers.
If n is odd,
th
n roots of
perfect nth powers
If n is even,
n
Product rule for nth roots
th
n
Quotient rule for n roots
n
an  a
n
an  a
ab  n a
n
b
a na

, b0
b nb
Properties of Radicals
1.
n
x x
n
2.
n
xy  x y
n
Page 12
n
3.
n
x

y
n
n
x
y
Transition to College Math
Review Notes
Examples:
Change from rational exponent form to radical form.
1
1. u 5  5 u
2
2. (6 x 2 y 5 ) 9  9 (6 x 2 y 5 ) 2  9 36 x 4 y10
3. (3xy)

3
5
1
 5 (3xy) 3 
27 x3 y 3
Change from radical form to rational exponent form.
5
4.
4
9u  (9u )
1
4
4
5.  7 (2 x)4  (2 x) 7
1
6. 3 x3  y 3  ( x3  y 3 ) 3
Simplify.
7.
7
6 2  12  4  3  2 3
8.
9.
(u 2  v 2 )7  u 2  v 2
3
x2 3 x2

8
2
18 x5 y 2 z 3  3x 2 | yz | 2 xz
10.
2
11.
4
1
27a3b3 4 3a5b3  4 81a8b6  3a 2b 4 b2  3a 2b(b 4 )  3a 2b(b 2 )  3a 2 | b | b
Page 13
Transition to College Math
12.
9
1
9
2
3
1
3
Review Notes
1
3 9
2
3
1
3
1
3
2
3
1
3
8 x y  8 x y  (2 ) x y  2 x y  3 2 x 2 y
6
1
3
2
1
3
11

1
1
1
4  4  4  4 3 2  4 6  (22 ) 6  2 3  3 2
3
13.
3
14. 6 2  2 2  8 2
15. 3 5 2 x 2 y 3  8 5 2 x 2 y 3  5 5 2 x 2 y 3
16. 5 3 mn2  3 mn  2 3 mn2  7 mn  3 3 mn2  4 mn
17.
3


6  4  18  4 3  3 2  4 3
18.

32
19.

y 2
20.

3

3  4  9  4 3  2 3  8  3  2 3  8  5  2 3

y 4 
x2  3 y 2



3
y2  4 y  2 y  8  y  2 y  8

x  3 y  3 x3  3 x 2 y  3 xy 2  3 y 3  x  3 x 2 y  3 xy 2  y
Page 14
Transition to College Math
Review Notes
Eliminating a radical from a denominator is referred to as rationalizing the denominator.
Examples:
6
6
2x 6 2x 3 2x




2x
x
2x
2x 2x
1.
x2

y3
2.
3.
4.
10 x3 10 x3 3 2 x 2 10 x3 3 2 x 2



 5x2 3 2 x2
3
3 2
3
2
2
x
4x
2 x 2x
3
8c

9d 5
5.
x 2
x  2 2 x  3 2 x2  3 x  4 x  6 2 x  x  6




4x  9
2 x 3 2 x 3 2 x 3
4 x2  9
6.
3 5

3 5
7. (Hint: Remember Difference of Cubes a3  b3  (a  b)(a 2  ab  b2 ) ).
1 3 y  3 y2
1 3 y  3 y2
1 3 y  3 y2
1
1




1 y
1  3 y 1  3 y 1  3 y  3 y 2 1  3 y  3 y 2  3 y  3 y 2  3 y3
Page 15
Transition to College Math
Review Notes
R.4 Polynomials: Basic Operations
Essential Question(s):
 How do you add, subtract, and multiply polynomials?
Definitions:
Polynomial
An algebraic expression involving only the operations of
addition, subtractions, multiplication, and raising to
natural number powers
Term
Part of a polynomial separated by  symbols
Degree
The highest exponent
Degree of the Term
The sum of the powers of the variables
Degree of a
Polynomial
The degree of the nonzero term with the highest degree in
the polynomial
Coefficient
The number in front of a variable
Leading Coefficient
The first number in an expression
Constant Term
The “Lone number” – without a variable
Descending Order
Highest to lowest
Like Terms
Polynomials that have exactly the same variable factors
raised to the same powers
Special Types
Monomial –
one term
Example: 8x 2 y
Binomial –
two terms
Example: 4 x  5
Trinomial –
three terms
Example: a 2  2ab  b 2
Page 16
Transition to College Math
Review Notes
Examples:
1. Which of the following are polynomials?
3x 2  2 x  1
x 3
x 2  2 xy  y 2
x 1
x2  2
2. Given the polynomial 3x5  6 x3  5 , what is the degree of the first term? The second
term? The whole polynomial?
5 ; 3; 5
3. Given the polynomial 6 x 4 y 2  3xy 3 , what is the degree of the first term? The second
term? The whole polynomial?
6; 4; 6
Remove parentheses and combine like terms.
4. 3(u 2  2v 2 )  (u 2  5v 2 )
3u 2  6v 2  u 2  5v 2
4u 2  v 2
5. (m3  3m2  m  1)  (2m3  m  3)
m3  3m2  m  1  2m3  m  3
m3  3m2  2m  4
6. ( x3  2)  [2 x3  (3x  4)]
x 3  2  [2 x 3  3 x  4]
x3  2  2 x3  3x  4
 x3  3x  2
7. Add 3x 4  2 x3  4 x 2 , x 3  2 x 2  5 x , and x 2  7 x  2 .
3x 4  2 x3  4 x 2  x3  2 x 2  5 x  x 2  7 x  2
3x 4  x3  5 x 2  2 x  2
8. Subtract 2 x 2  5 x  4 from 5 x 2  6 .
5 x 2  6  (2 x 2  5 x  4)
5x2  6  2 x2  5x  4
3x 2  5 x  10
Page 17
Transition to College Math
Review Notes
Multiplying Polynomials
Monomial x Polynomial:
Distribute
2(2 x 2  4 x  9)
4 x 2  8 x  18
Binomial x Binomial:
First
Outer
Inner
Last
(2 x  3 y )(5 x  2 y )
10 x 2  4 xy  15 xy  6 y 2
10 x 2  11xy  6 y 2
Polynomial x Polynomial:
Clamshell Method
(2 x  3)(3 x 2  2 x  3)
6 x3  4 x 2  6 x  9 x 2  6 x  9
6 x3  13 x 2  12 x  9
Special Products:
a 2  b 2  (a  b)(a  b)
25 x 2  81  (5 x  9)(5 x  9)
a 2  2ab  b 2  (a  b) 2
4 x 2  12 x  9  (2 x  3) 2
a 2  2ab  b 2  (a  b) 2
49 y 2  28 y  4  (7 y  2) 2
Examples:
Multiply.
1. (2 x  3)(2x 2  3x  2)
4 x3  6 x 2  4 x  6 x 2  9 x  6
4 x3  13x  6
2. (4u  3v)(2u  v)
8u 2  4uv  6uv  3v 2
8u 2  2uv  3v 2
3. (2 xy  3)(2 xy  3)
Page 18
Transition to College Math
Review Notes
4 x2 y 2  9
4. (m  4n)(m  4n)
m2  16n 2
5. (2u  3v)2
4u 2  12uv  9v 2
6. (6 x  y)2
36 x 2  12 xy  y 2
Perform the indicated operations and simplify.
7. 2t  {7  2[t  t (4  t )]}
2t  {7  2[t  4t  t 2 ]}
2t  {7  2[3t  t 2 ]}
2t  {7  6t  2t 2 }
2t  7  6t  2t 2
2t 2  4t  7
8. (u  3v)2  (2u  v)(2u  v)
u 2  6uv  9v 2  (4u 2  v 2 )
u 2  6uv  9v 2  4u 2  v 2
3u 2  6uv  10v 2
9. (4 x  y )3
(4 x  y )(16 x 2  8 xy  y 2 )
64 x 3  32 x 2 y  4 xy 2  16 x 2 y  8 xy 2  y 3
64 x 3  48 x 2 y  12 xy 2  y 3
11. A plastic water pipe is 200 inches long, 2 inches thick, and has an outer radius of x inches.
Write an algebraic expression in terms of x that represents the volume of the plastic used to
construct the pipe. Simplify the expression.
V   x 2 (200)   ( x  2) 2 (200)
V  200 x 2  200 ( x 2  4 x  4)
V  200 x 2  200 x 2  800 x  800
V  800 x  800
R.5 Polynomials: Factoring
Page 19
Transition to College Math
Review Notes
Essential Question(s):
 How do you factor polynomials?
Definitions:
Prime Number
An integer greater than one having no integral factors other than itself and
one.
Composite Number
An integer greater than one that is not prime
Methods to Factor Polynomials
1. GCF: ALWAYS FACTOR GCF FIRST!
a. Find the GCF of each term
b. Divide each term by the GCF.
Ex. 3x3  15x2  3x2 ( x  5)
2. Difference of Square: a 2  b2  (a  b)(a  b)
Two
Terms
Ex. 4 x 2  9  (2 x  3)(2 x  3)
3. Sum & Difference of Cubes:
a3  b3  (a  b)(a 2  ab  b 2 )
a3  b3  (a  b)(a 2  ab  b 2 )
Ex. 8x3  y3  (2 x  y)(4 x 2  2 xy  y 2 )
4. Perfect Square Trinomial:
a 2  2ab  b2  (a  b)2
a 2  2ab  b2  (a  b)2
Ex. n2  4n  4  (n  2)2
Three
Terms
5. AC Method to Factor Quadratic Polynomial ax 2  bx  c :
a. Multiply ac.
b. Find factors of product of ac that has a sum equal to b.
c. Rewrite bx term as the sum of factors found from Step 2.
d. Factor by grouping.
Ex.
2 x2  5x  3
2(3)  6
1, 6
 1, 6
2, 3
 2,3
1, 6 product is  6 and sum is 5
Page 20
2 x 2  1x  6 x  3
x(2 x  1)  3(2 x  1)
(2 x  1)( x  3)
Transition to College Math
Four
Terms
Review Notes
6. Grouping:
a. Rearrange terms into 2 subgroups and factor.
Ex.
x2  4x  4  4 y 2
xy  3 y  2 x  6
( x 2  4 x  4)  4 y 2
y ( x  3)  2( x  3)
( x  2) 2  4 y 2 Factor as difference of squares.
( x  3)( y  2)
( x  2  2 y )( x  2  2 y )
Examples: Factor completely.
1. 3x3 y  6 x2 y 2  3xy3
3xy( x2  2 xy  y 2 )
2. 3 y (2 y  5)  2(2 y  5)
(2 y  5)(3 y  2)
3. 4(2 x  5)(3x  1)2  6(2 x  5)2 (3x  1)
2(2 x  5)(3 x  1)[2(3x  1)  3(2 x  5)]
2(2 x  5)(3 x  1)[6 x  2  6 x  15]
2(2 x  5)(3 x  1)(12 x  17)
4. 2 x 2  6 x  5 x  15
2 x( x  3)  5( x  3)
( x  3)(2 x  5)
5. 2 pr  ps  6qr  3qs
2 p(r  s )  3r (r  s )
(r  s )(2 p  3r )
6. 2 x 2  7 xy  4 y 2
2 x 2  8 xy  xy  4 y 2
2 x( x  4 y )  y ( x  4 y )
( x  4 y )(2 x  y )
7. 4 x 2  15xy  4 y 2
4 x 2  16 xy  xy  4 y 2
4 x( x  4 y )  y ( x  4 y )
( x  4 y )(4 x  y )
Page 21
Transition to College Math
Review Notes
8. 4m2  12mn  9n 2
(2m  3n)2
9. x 2  16 y 2
( x  4 y )( x  4 y )
10. z 3  1
( z  1)( z 2  z  1)
11. m 3  n3
(m  n)(m2  mn  n2 )
12. 3x3  48 x
3x( x 2  16)
3x( x  4)( x  4)
13. x 2  y 2  4 y  4
x 2  ( y 2  4 y  4)
x 2  ( y  2) 2
[ x  ( y  2)][ x  ( y  2)]
( x  y  2)( x  y  2)
14. 3u 4  3u 3v  9u 2 v 2
3u 2 (u 2  uv  3v 2 )
15. 3m4  24mn3
3m(m3  8n3 )
3m(m  2n)(m2  2mn  4n 2 )
16. 3 x 4  5 x 2  2
3x 4  3x 2  2 x 2  2
3 x 2 ( x 2  1)  2( x 2  1)
( x 2  1)(3 x 2  2)
( x  1)( x  1)(3 x 2  2)
Page 22
Transition to College Math
Review Notes
R.6 Rational Expressions
Essential Question(s):
 How do you simplify and add, subtract, multiply, and divide rational expressions?
 How do you simplify complex fractions?
Remember, Rational means… fractional
Rational expression – the quotient of two polynomials
Note: An Expression is a collection of terms or an incomplete mathematical thought while an
equation is a completed thought having an answer.
Note: Expressions can only be simplified while equations can be solved.
I.
The Domain of a Rational Expression
The domain of an algebraic expression is…
all real numbers except those that make the denominator equal to zero
Example: Find the domain.
43
x 9
2
Look at the denominator. What values make x 2  9  0 ?
( x  3)( x  3)  0
x  3
 OR 
x2  9
x  3
So, the domain is all real numbers except 3 or {x | x  3}
II.
Simplifying, Multiplying, and Dividing an Expression
Simplify. Give restrictions
Example:
Simplify.
7 y 2  11y  6
y3  y 2  6 y
Example:
(7 y  3) ( y  2)
y ( y  2) ( y  3)
7y 3
, y  0,  2, 3
y ( y  3)
x2  x
y 
 2 

2y  y
3x 
 1 
x2
  
2y
 3y 
x2
 3y 
  
2y  1 
3x 2
2
Page 23
Transition to College Math
III.
Review Notes
Adding and Subtracting Rational Expressions
Use LCD
Example:
Find LCD: ( w  1)( w  1)
w2  1 w  1

w2  1 w  1
w2  1  w  1  w  1 



w2  1  w  1  w  1 
w2  1 w2  2 w  1

w2  1
w2  1
2 w2  2 w  2
( w  1)( w  1)
2( w2  w  1)
( w  1)( w  1)
Example:
5
12

w 1 1 w
5
12

w 1 w 1
7
w 1
IV.
Complex Rational Expressions
Procedures:
1. Find the LCD for all denominators in the complex rational expression
2. Multiply by a form of “1”
multiply the expression by
3. The result should be a simple rational expression.
Page 24
LCD
.
LCD
Transition to College Math
Example:
Find LCD:
Review Notes
m mn
n
n
n
m
m
m
n

n
n
m
mn  m
m

n
mn  n
m2n  m2

mn 2  n 2
m 2 (n  1)
 2
n (m  1)
m
m

m



n   mn 


n   mn 

n 
m

m2n  m2

mn 2  n 2
m 2 (n  1)
 2
n (m  1)
OR
5  w1
Example:
12  w1
1

5   w
w  
 
1   w 

12



w

5w  1

12 w  1
OR
Page 25
1

5  
w
 
1

12  
w

5w  1
w

w
12 w  1
5w  1

12 w  1