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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.

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Presentation on theme: "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."— Presentation transcript:

1 College Algebra Chapter R

2 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

3 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”

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

5 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”

6 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

7 College Algebra R.1 Definition of Absolute Value:

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

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

10 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

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

12 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

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

14 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

15 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

16 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

17 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 = x Zero is the identity for addition x*1=x 1*x=x one is the identity for multiplication

18 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 = 0x + (-x) = 0 -x is the additive inverse for any real number x is the multiplicative inverse for any real number

19 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

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

21 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?

22 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:

23 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:

24 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:

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

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

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

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

29 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

30 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

31 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials ADDING AND SUBTRACTING POLYNOMIALS

32 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials ADDING AND SUBTRACTING POLYNOMIALS

33 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS

34 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS

35 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS

36 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

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

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

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

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

41 College Algebra R.4 Factoring Polynomials Factoring quadratic polynomials

42 College Algebra R.4 Factoring Polynomials Factoring quadratic polynomials

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

44 College Algebra R.4 Factoring Polynomials Factoring Special Forms

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

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

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

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

49 College Algebra R.4 Factoring Polynomials Classwork

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

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

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

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

54 College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS 1.Factor all numerators and denominators 2.Reduce common factors 3.Multiply (numerator x numerator) (denominator x denominator)

55 College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS 1.Factor all numerators and denominators 2.Reduce common factors 3.Multiply (numerator x numerator) (denominator x denominator)

56 College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS 1.Factor all numerators and denominators 2.Reduce common factors 3.Multiply (numerator x numerator) (denominator x denominator)

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

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

59 College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS 1.Find LCD of all denominators 2.Build equivalent expressions 3.Add or subtract numerators 4.Write the result in lowest terms

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

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

62 College Algebra R.5 Rational Expressions Simplifying Compound Fractions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

78 Review Simplify College Algebra R

79 Review Simplify College Algebra R

80 Review Simplify College Algebra R

81 Review Factor College Algebra R

82 Homework pg College Algebra R


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