# College Algebra Chapter R.

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College Algebra Chapter R

Sets of Numbers: N W Z Q H R Natural Numbers Whole Numbers Integers
College Algebra R.1 Sets of Numbers: N W Z Q H R Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers 1,2,3,… 0,1,2,3,… …,-2,-1,0,1,2,… Fractions Cannot be fractions Combination of Q and H

Set Notation: Braces { } used to group members/elements of a set
College Algebra R.1 Set Notation: Braces { } used to group members/elements of a set Empty or null set designated by empty braces or The 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”

List the natural numbers less than 6
College Algebra R.1 List the natural numbers less than 6 List the natural numbers less than 1 {1,2,3,4,5} { } Identify each of the following statements as true or false True False; 2.2 is not a whole number

Inequality symbols Greater than > Less than <
College Algebra R.1 Inequality symbols Greater than > Less than < Strict vs. Nonstrict inequalities strict inequalities means the endpoints are included in the relation aka “less than or equal to” and “greater than or equal to”

Absolute Value of a Real Number
College Algebra R.1 Absolute Value of a Real Number The 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

Definition of Absolute Value:
College Algebra R.1 Definition of Absolute Value:

Division and Zero The quotient of Zero and any real number n is zero
College Algebra R.1 Division and Zero The quotient of Zero and any real number n is zero Are undefined

Square Roots Cube Roots This also means that This also means that
College Algebra R.1 Square Roots Cube Roots This also means that This also means that

Order of Operations 1 grouping symbols 2 exponents and roots
College Algebra R.1 Order of Operations 1 grouping symbols 2 exponents and roots 3 division and multiplication 4 subtraction and addition 23,020.89

College Algebra R.1 Homework pg 10 (1-100)

Algebraic Expressions Terminology
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

Decomposition of Rational Terms
College Algebra R.2 Decomposition of Rational Terms For any rational term

Evaluating a Mathematical Expression
College Algebra R.2 Evaluating a Mathematical Expression 1 replace each variable with an open Parenthesis ( ). 2 substitute the given replacements for each variable 3 simplify using order of operations

Properties of Real Numbers
College Algebra R.2 Properties of Real Numbers THE COMMUTATIVE PROPERTIES Given that a and b represent real numbers: ADDITION: a + b = b + a Addends can be combined in any order without changing the sum. MULTIPLICATION: a*b=b*a Factors can be multiplied in any order without changing the product

Properties of Real Numbers
College Algebra R.2 Properties of Real Numbers THE 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

Properties of Real Numbers
College Algebra R.2 Properties of Real Numbers THE ADDITIVE AND MULTIPLICATIVE IDENTITIES Given that x is a real number: x + 0 = x x = x Zero is the identity for addition x*1=x *x=x one is the identity for multiplication

Properties of Real Numbers
College Algebra R.2 Properties of Real Numbers THE ADDITIVE AND MULTIPLICATIVE INVERSES given that a, b, and x represent real numbers where a, b = 0 -x + x = 0 x + (-x) = 0 -x is the additive inverse for any real number x is the multiplicative inverse for any real number

Properties of Real Numbers
College Algebra R.2 Properties of Real Numbers THE DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION Given that a, b, and c represent real numbers: a(b+c) = ab + ac A factor outside a sum can be distributed to each addend in the sum ab + ac = a(b+c) A factor common to each addend in a sum can be “undistributed” and written outside a group

College Algebra R.2 Homework pg 19 (1-98)

An exponent tells us how many times the base b is used as a factor
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials EXPONENTIAL NOTATION An exponent tells us how many times the base b is used as a factor What would look like?

PRODUCT PROPERTY OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT PROPERTY OF EXPONENTS For any base b and positive integers m and n: POWER PROPERTY OF EXPONENTS For any base b and positive integers m and n:

PRODUCT TO A POWER PROPERTY
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: QUOTIENT TO A POWER PROPERTY For any bases a and b, and positive integers m, n, and p:

QUOTIENT PROPERTY OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials QUOTIENT PROPERTY OF EXPONENTS For any base b and integer exponents m and n: ZERO EXPONENT PROPERTY For any base b:

PROPERTY OF NEGATIVE EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PROPERTY OF NEGATIVE EXPONENTS

PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PROPERTIES OF EXPONENTS

PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PROPERTIES OF EXPONENTS

A number written in scientific notation has the form
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials SCIENTIFIC NOTATION A number written in scientific notation has the form

College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATION Write in scientific notation Write in standard notation Simplify and write in scientific notation

POLYNOMIAL TERMINOLOGY
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 Show horizontal and vertical method

College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials ADDING AND SUBTRACTING POLYNOMIALS Show horizontal and vertical method

PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS Show FOIL and box method

PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS Show FOIL and box method

PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS Show FOIL and box method

SPECIAL POLYNOMIAL PRODUCTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials SPECIAL POLYNOMIAL PRODUCTS Binomial conjugates For any given binomial, its conjugate is found by using the same two terms with the opposite sign between them Product of a binomial and its conjugate Square of a binomial

College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
Homework pg 31 (1-140)

Greatest Common Factor
College Algebra R.4 Factoring Polynomials Greatest Common Factor Largest factor common to all terms in a polynomial

Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials Common Binomial Factors and Factoring by Grouping

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

Factoring Special Forms
College Algebra R.4 Factoring Polynomials Factoring Special Forms Difference of 2 perfect squares Perfect square trinomials

Factoring Special Forms
College Algebra R.4 Factoring Polynomials Factoring Special Forms

Factoring Special Forms
College Algebra R.4 Factoring Polynomials Factoring Special Forms Sum or difference of two cubes

Factoring Special Forms
College Algebra R.4 Factoring Polynomials Factoring Special Forms Sum or difference of two cubes

U-substitution or placeholder substitution
College Algebra R.4 Factoring Polynomials U-substitution or placeholder substitution

Factoring Polynomials
College Algebra R.4 Factoring Polynomials Factoring flow chart on page 41 Factoring Polynomials GCF Number of Terms Two Three Four Grouping Trinomials (a=1) Difference of Squares Trinomials (a=1) Advanced Methods (4.2) Difference of Cubes Sum of Cubes

College Algebra R.4 Factoring Polynomials
Classwork

College Algebra R.4 Factoring Polynomials
Homework pg 41 (1-60)

FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS If P, Q, and R are polynomials, where Q or R = 0 then,

FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS Reduce to lowest terms

FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS Simplify each and state the excluded values.

MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS Factor all numerators and denominators Reduce common factors Multiply (numerator x numerator) (denominator x denominator)

MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS Factor all numerators and denominators Reduce common factors Multiply (numerator x numerator) (denominator x denominator)

MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS Factor all numerators and denominators Reduce common factors Multiply (numerator x numerator) (denominator x denominator)

DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions DIVISION OF RATAIONAL EXPRESSIONS Invert divisor and multiply as before

DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions DIVISION OF RATAIONAL EXPRESSIONS

ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS Find LCD of all denominators Build equivalent expressions Add or subtract numerators Write the result in lowest terms

ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS

ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS

Simplifying Compound Fractions
College Algebra R.5 Rational Expressions Simplifying Compound Fractions

College Algebra R.5 Rational Expressions
Homework pg 50 (1-84)

The square root of For any real number The cube root of
College Algebra R.6 Radicals and Rational Exponents The square root of For any real number The cube root of

The nth root of For any real number a,
College Algebra R.6 Radicals and Rational Exponents The nth root of For any real number a,

If is a real number with and , then
College Algebra R.6 Radicals and Rational Exponents RATIONAL EXPONENTS If is a real number with and , then

For m, with m and n relatively prime and
College Algebra R.6 Radicals and Rational Exponents RATIONAL EXPONENTS For m, with m and n relatively prime and

College Algebra R.6 Radicals and Rational Exponents
Homework pg 64 (1-62)

Review True False College Algebra R State true or false.
Simplify by combining like terms

College Algebra R Review Simplify

College Algebra R Review Simplify

College Algebra R Review Simplify

College Algebra R Review Factor

College Algebra R Homework pg

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