College AlgebraChapter R
College Algebra R.1
Sets of Numbers:
NWZQHR
Natural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers
1,2,3,…0,1,2,3,……,-2,-1,0,1,2,…FractionsCannot be fractionsCombination of Q and H
College Algebra R.1
Set Notation:Braces { } used to group members/elementsof a set
Empty or null set designated by empty braces orThe symbol Ø
To show membership in a set use read“is an element of” or “belongs to”
Other notation:“is a subset of”“is not an element of”
College Algebra R.1
List the natural numbers less than 6
List the natural numbers less than 1
Identify each of the following statements astrue or false
{1,2,3,4,5}
{ }
WN W}5.24.2,3.2,2.2{
True
False; 2.2 is not a whole number
College Algebra R.1
Inequality symbols
Greater than >
Less than <
Strict vs. Nonstrict inequalitiesstrict inequalities means the endpointsare included in the relation aka “less thanor equal to” and “greater than or equal to”
College Algebra R.1
Absolute Value of a Real NumberThe absolute value of a real number a, denoted |a|, is the undirected distance between a and 0 on the number line: |a|>0
9 - |7 – 15| = ? 1
College Algebra R.1
Definition of Absolute Value:
0
0
xifx
xifxx
College Algebra R.1
Division and Zero
The quotient of Zero and any real number n is zero 0n
nandn
000
00
nandn Are undefined
College Algebra R.1
Square Roots
0
A
ABBifBA
This also means that
AA
AAA
2
Cube Roots
RA
ABBBifBA
3
This also means that
AA
AAAA
3
3
333
College Algebra R.1
Order of Operations
1 grouping symbols2 exponents and roots3 division and multiplication4 subtraction and addition
1512
12
075.017500
23,020.89
College Algebra R.1
Homework pg 10 (1-100)
College Algebra R.2
Algebraic Expressions Terminology
Algebraic Term
Constant
Variable Term
Coefficient
Algebraic Expression
A collection of factors
Single nonvariable number
Any term that contains a variable
Constant factor of a variable term
Sum or difference of algebraic terms
College Algebra R.2
Decomposition of Rational Terms
For any rational term 0BB
A
AB
A
BB
A
1
1
1
xx
27
3
xx 237
1
College Algebra R.2
Evaluating a Mathematical Expression1 replace each variable with an openParenthesis ( ).2 substitute the given replacements for eachvariable3 simplify using order of operations
822 2 xx
822 2 82222 2
848
College Algebra R.2
Properties of Real NumbersTHE COMMUTATIVE PROPERTIES
Given that a and b represent real numbers:
ADDITION: a + b = b + aAddends can be combined in any order withoutchanging the sum.
MULTIPLICATION: a*b=b*aFactors can be multiplied in any orderwithout changing the product
College Algebra R.2
Properties of Real NumbersTHE ASSOCIATIVE PROPERTIES
Given that a, b and c represent real numbers:
ADDITION: (a + b) + c = a + (b + c)Addends can be regrouped.
MULTIPLICATION: (a*b)*c=a*(b*c)Factors can be regrouped
College Algebra R.2
Properties of Real Numbers
THE ADDITIVE AND MULTIPLICATIVE IDENTITIES
Given that x is a real number:
x + 0 = x 0 + x = xZero is the identity for addition
x*1=x 1*x=xone is the identity for multiplication
College Algebra R.2
Properties of Real Numbers
THE ADDITIVE AND MULTIPLICATIVE INVERSESgiven that a, b, and x represent real numberswhere a, b = 0-x + x = 0 x + (-x) = 0-x is the additive inverse for any real number x
11 b
a
a
b
a
b
b
a
is the multiplicative inverse for any realnumberb
a
a
b
College Algebra R.2
Properties of Real NumbersTHE DISTRIBUTIVE PROPERTY OFMULTIPLICATION OVER ADDITION
Given that a, b, and c represent real numbers:
a(b+c) = ab + acA factor outside a sum can be distributedto each addend in the sum
ab + ac = a(b+c)A factor common to each addend in a sum canbe “undistributed” and written outside a group
College Algebra R.2
Homework pg 19 (1-98)
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
EXPONENTIAL NOTATION
An exponent tells us how many times the base bis used as a factor
n
timesntimesn
n bbbbbandbbbbb
32xWhat would look like? 222 xxx
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT PROPERTY OF EXPONENTS
For any base b and positive integers m and n:nmnm bbb 52 55 103 xx 84 nn
POWER PROPERTY OF EXPONENTS
For any base b and positive integers m and n:
nmnm bb 23x
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT TO A POWER PROPERTY
For any bases a and b, and positive integers m, n, and p:
npmppnm baba 323xQUOTIENT TO A POWER PROPERTY
np
mpp
n
m
b
a
b
a
For any bases a and b, and positive integers m, n, and p:23
2
5
b
a
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
QUOTIENT PROPERTY OF EXPONENTS
For any base b and integer exponents m and n:
0, bbb
b nmn
m
ZERO EXPONENT PROPERTY
For any base b:
0,10 bifb
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTY OF NEGATIVE EXPONENTSnnn
nn
n
a
b
b
ab
bb
b
1
11
1
?2
2
2
3
b
a ?6323232
khhk
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS
?6
9
x
x
?2
25
nm
nm
?9
922
33
s
s
qp
qp
?4
2
3
c
ba
3x
nm3
pq
84
12
cb
a
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATION
A number written in scientific notation has the formkN 10
egeraniskandN int101
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATIONWrite in scientific notation
Write in standard notation
Simplify and write in scientific notation
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
POLYNOMIAL TERMINOLOGY
Monomial
Degree
Polynomial
Degree of a polynomial
Binomial
Trinomial
A term using only whole number exponents
The same as the exponent on the variable
A monomial or any sum or difference of monomial terms
Is the largest exponent occurring on any variable
A polynomial with two terms
A polynomial with three terms
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
?82395 233 xxxxx
?82395 233 xxxxx
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
?2312 xx
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SPECIAL POLYNOMIAL PRODUCTS
Binomial conjugatesFor any given binomial, its conjugate is found by using the same two termswith the opposite sign between them
7x 7xProduct of a binomial and its conjugate
?77 xx ? BABA
Square of a binomial
?7 2 x ?2 BA
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
Homework pg 31 (1-140)
College Algebra R.4 Factoring Polynomials
Greatest Common FactorLargest factor common to all terms in a polynomial
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Difference of 2 perfect squares
BABABA 22
Perfect square trinomials
222 2 BABABA
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Sum or difference of two cubes
2233
2233
BABABABA
BABABABA
College Algebra R.4 Factoring Polynomials
Factoring Special FormsSum or difference of two cubes
1258 3 x 6427 3 r
College Algebra R.4 Factoring Polynomials
U-substitution or placeholder substitution
3222 222 xxxx
College Algebra R.4 Factoring Polynomials
Factoring flow chart on page 41Factoring
Polynomials GCF
Number of Terms
FourThreeTwo
Difference of Squares
Difference of Cubes
Sum of Cubes
Trinomials (a=1)
Trinomials (a=1)
Advanced Methods (4.2)
Grouping
College Algebra R.4 Factoring Polynomials
Classwork
4236 2 xx 9295 323 zzxxyzy
910 nn 58 mm
337 rr 973 bb
2vu yxyx 33
dcxa 26 2 22338 kcxa
College Algebra R.4 Factoring Polynomials
Homework pg 41 (1-60)
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONSIf P, Q, and R are polynomials, where Q or R = 0 then,
RQ
RP
Q
Pand
Q
P
RQ
RP
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Reduce to lowest terms
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Simplify each and state the excluded values.
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
QS
PR
S
R
Q
P
1. Factor all numerators and denominators2. Reduce common factors3. Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS1. Factor all numerators and denominators2. Reduce common factors3. Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS1. Factor all numerators and denominators2. Reduce common factors3. Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS
QR
PS
R
S
Q
P
S
R
Q
P
Invert divisor and multiply as before
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OFRATAIONAL EXPRESSIONS
1. Find LCD of all denominators2. Build equivalent expressions3. Add or subtract numerators4. Write the result in lowest terms
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OFRATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OFRATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
Simplifying Compound Fractions
?
31
43
23
32
2
mm
m
College Algebra R.5 Rational Expressions
Homework pg 50 (1-84)
College Algebra R.6 Radicals and Rational Exponents
The square root of
For any real number
The cube root of
For any real number
22 : aa
aaa 2,
3 33 : aa
aaa 3 3,
College Algebra R.6 Radicals and Rational Exponents
The nth root of
For any real number a,
n nn aa :
oddisawhenaa
evenisawhenaa
n n
n n
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
If is a real number with and , thenZn 2nn R
nnn RRR1
1
?27
83
3
w
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
For m, with m and n relatively prime and Zn 2n
n mn
m
mnn
m
RR
RR
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
nnnnnn ABBAandBAAB
Product property of radicals
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Product property of radicals
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
nn
n
n
n
n
B
A
B
Aand
B
A
B
A
Quotient property of radicals
?125
813
3
x
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical ExpressionsRationalizing the denominator
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical ExpressionsRationalizing the denominator
College Algebra R.6 Radicals and Rational Exponents
Homework pg 64 (1-62)
College Algebra R
Review
State true or false.
RH
Q2
Simplify by combining like terms
xxxxx 324 2
True
False
26 xx
Review
Simplify
342232 baa 12124 ba
zx81
276025 2 xx
964 24 aaa
College Algebra R
Review
Simplify
6133 23 xxx
23
3
mm
m
29
2
b
b
College Algebra R
Review
104
62
rr
r
Simplify
612
751048 pp
College Algebra R
ReviewFactor
2243 2439 pqqqm
3926 nn
5437 2 xx
College Algebra R
Homework pg 68 1-20
College Algebra R