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College Algebra Chapter R

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**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

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**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”

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**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

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**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”

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**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

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**Definition of Absolute Value:**

College Algebra R.1 Definition of Absolute Value:

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**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

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**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

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**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

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

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**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

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**Decomposition of Rational Terms**

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

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**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

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**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

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**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

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**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

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**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

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**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

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

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**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?

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**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:

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**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:

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**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:

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**PROPERTY OF NEGATIVE EXPONENTS**

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

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**PROPERTIES OF EXPONENTS**

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

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**PROPERTIES OF EXPONENTS**

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

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**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

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**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

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**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

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**ADDING AND SUBTRACTING POLYNOMIALS**

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

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**ADDING AND SUBTRACTING POLYNOMIALS**

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

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**PRODUCT OF TWO POLYNOMIALS**

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

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**PRODUCT OF TWO POLYNOMIALS**

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

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**PRODUCT OF TWO POLYNOMIALS**

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

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**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

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**College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials**

Homework pg 31 (1-140)

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**Greatest Common Factor**

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

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**Common Binomial Factors and Factoring by Grouping**

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

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**Common Binomial Factors and Factoring by Grouping**

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

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**Factoring quadratic polynomials**

College Algebra R.4 Factoring Polynomials Factoring quadratic polynomials

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**Factoring quadratic polynomials**

College Algebra R.4 Factoring Polynomials Factoring quadratic polynomials

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**Factoring Special Forms**

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

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**Factoring Special Forms**

College Algebra R.4 Factoring Polynomials Factoring Special Forms

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**Factoring Special Forms**

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

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**Factoring Special Forms**

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

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**U-substitution or placeholder substitution**

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

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**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

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**College Algebra R.4 Factoring Polynomials**

Classwork

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**College Algebra R.4 Factoring Polynomials**

Homework pg 41 (1-60)

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**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,

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**FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS**

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

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**FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS**

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

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**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)

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**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)

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**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)

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**DIVISION OF RATAIONAL EXPRESSIONS**

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

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**DIVISION OF RATAIONAL EXPRESSIONS**

College Algebra R.5 Rational Expressions DIVISION OF RATAIONAL EXPRESSIONS

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**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

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**ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS**

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

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**ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS**

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

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**Simplifying Compound Fractions**

College Algebra R.5 Rational Expressions Simplifying Compound Fractions

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**College Algebra R.5 Rational Expressions**

Homework pg 50 (1-84)

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**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

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**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,

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**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

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**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

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Product property of radicals

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Product property of radicals

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Quotient property of radicals

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions Rationalizing the denominator

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**PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS**

College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions Rationalizing the denominator

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**College Algebra R.6 Radicals and Rational Exponents**

Homework pg 64 (1-62)

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**Review True False College Algebra R State true or false.**

Simplify by combining like terms

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College Algebra R Review Simplify

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College Algebra R Review Simplify

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College Algebra R Review Simplify

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College Algebra R Review Factor

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College Algebra R Homework pg

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