College Algebra Chapter R. College Algebra R.1 Sets of Numbers: N W Z Q H R Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers

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


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


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


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


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


PROPERTY OF NEGATIVE EXPONENTSnnn

nn

n

a

b

b

ab

bb

b

1

11

1

?2

2

2

3

b

a ?6323232

khhk


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


PROPERTIES OF EXPONENTS


SCIENTIFIC NOTATION

A number written in scientific notation has the formkN 10

egeraniskandN int101


SCIENTIFIC NOTATIONWrite in scientific notation

Write in standard notation

Simplify and write in scientific notation


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


ADDING AND SUBTRACTING POLYNOMIALS


ADDING AND SUBTRACTING POLYNOMIALS

?82395 233 xxxxx

?82395 233 xxxxx


PRODUCT OF TWO POLYNOMIALS

?2312 xx






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.4 Factoring Polynomials

Greatest Common FactorLargest factor common to all terms in a polynomial


Common Binomial Factors and Factoring by Grouping


Common Binomial Factors and Factoring by Grouping


Factoring quadratic polynomials


Factoring quadratic polynomials


Factoring Special Forms

Difference of 2 perfect squares

BABABA 22

Perfect square trinomials

222 2 BABABA





Sum or difference of two cubes

2233

2233

BABABABA

BABABABA


Factoring Special FormsSum or difference of two cubes

1258 3 x 6427 3 r


U-substitution or placeholder substitution

3222 222 xxxx


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


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.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


FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS

Reduce to lowest terms


FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS

Simplify each and state the excluded values.


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)


MULTIPLYING RATAIONAL EXPRESSIONS1. Factor all numerators and denominators2. Reduce common factors3. Multiply (numerator x numerator)



MULTIPLYING RATAIONAL EXPRESSIONS1. Factor all numerators and denominators2. Reduce common factors3. Multiply (numerator x numerator)



DIVISION OF RATAIONAL EXPRESSIONS

QR

PS

R

S

Q

P

S

R

Q

P

Invert divisor and multiply as before


DIVISION OF RATAIONAL 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






Simplifying Compound Fractions

?

31

43

23

32

2

mm

m



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,


The nth root of

For any real number a,

n nn aa :

oddisawhenaa

evenisawhenaa

n n

n n


RATIONAL EXPONENTS

If is a real number with and , thenZn 2nn R

nnn RRR1

1

?27

83

3

w


RATIONAL EXPONENTS

For m, with m and n relatively prime and Zn 2n

n mn

m

mnn

m

RR

RR


PROPERTIES OF RADICALS ANDSIMPLIFYING RADICAL EXPRESSIONS

nnnnnn ABBAandBAAB

Product property of radicals



Product property of radicals



nn

n

n

n

n

B

A

B

Aand

B

A

B

A

Quotient property of radicals

?125

813

3

x



Operations on Radical Expressions









Operations on Radical ExpressionsRationalizing the denominator



Operations on Radical ExpressionsRationalizing the denominator



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

Documents

College Algebra Chapter R. College Algebra R.1 Sets of Numbers: N W Z Q H R Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers