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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 Real Numbers 1,2,3,… 0,1,2,3,… …,-2,-1,0,1,2,… Fractions Cannot be fractions Combination of Q and H 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” College Algebra R.1 List the natural numbers less than 6 {1,2,3,4,5} List the natural numbers less than 1 {} Identify each of the following statements as true or false N W True {2.2, 2.3, 2.4 2.5} W False; 2.2 is not a whole number 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” 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 College Algebra R.1 Definition of Absolute Value: x if x 0 x x if x 0 College Algebra R.1 Division and Zero The quotient of Zero and any real number n is zero n 0 0 0 n 0 and n 0 and n 0 n Are undefined College Algebra R.1 Square Roots Cube Roots A B if B B A A 0 A B if B B B A A R This also means that This also means that A A A A A 2 3 3 A 3 A 3 A A A A 3 3 College Algebra R.1 Order of Operations 1 2 3 4 grouping symbols exponents and roots division and multiplication subtraction and addition 1215 0.075 75001 12 23,020.89 College Algebra R.1 Homework pg 10 (1-100) College Algebra R.2 Algebraic Expressions Terminology Algebraic Term A collection of factors Constant Single nonvariable number Variable Term Any term that contains a variable Coefficient Constant factor of a variable term Algebraic Expression Sum or difference of algebraic terms College Algebra R.2 Decomposition of Rational Terms For any rational term A B 0 B A 1 A 1 A B B 1 B x3 2x 7 1 x 3 2x 7 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 2x2 2x 8 2 2 2 8 22 22 8 848 2 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 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 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 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 a b 1 b a b a 1 a b a b is the multiplicative inverse for any real b number a 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) 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 b n b b b b and b b b b bn n times What would x 23 look like? n times x 2x 2x 2 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT PROPERTY OF EXPONENTS For any base b and positive integers m and n: b b b m n m n 52 55 x 3 x10 n 4 n 8 POWER PROPERTY OF EXPONENTS For any base b and positive integers m and n: b m n b mn x 3 2 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: a m b n p a b mp np 3x 2 3 QUOTIENT TO A POWER PROPERTY For any bases a and b, and positive integers m, n, and p: a n b m p a mp np b 5a 2b 3 2 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials QUOTIENT PROPERTY OF EXPONENTS For any base b and integer exponents m and n: bm mn b , b0 n b ZERO EXPONENT PROPERTY For any base b: b 1, if b 0 0 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PROPERTY OF NEGATIVE EXPONENTS n n b 1 n 1 b 2a 2 b 3 1 b n b 1 2 ? a b 3hk 6h 2 3 2 k n b a 3 2 ? n College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PROPERTIES OF EXPONENTS x9 ? 6 x x m5 n 2 ? 2 mn 3 mn 9 p q 9 p q pq 3 3 s 2 2 s 4 ? a b 2 ? c 3 12 3 a 4 8 bc 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 form N 10 k 1 N 10 and k is an int eger 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 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials POLYNOMIAL TERMINOLOGY Monomial A term using only whole number exponents Degree The same as the exponent on the variable Polynomial A monomial or any sum or difference of monomial terms Degree of a polynomial Is the largest exponent occurring on any variable Binomial A polynomial with two terms Trinomial 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 x x 3 5x 9 x 3x 2x 8 ? 3 5x 9 x3 3x 2 2x 8 ? 3 2 College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials PRODUCT OF TWO POLYNOMIALS 2x 13x 2 ? 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 conjugates For any given binomial, its conjugate is found by using the same two terms with the opposite sign between them x7 x7 Product of a binomial and its conjugate x 7x 7 ? A B A B ? Square of a binomial x 7 2 ? A B 2 ? College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials Homework pg 31 (1-140) College Algebra R.4 Factoring Polynomials Greatest Common Factor Largest 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 A B A B A B 2 2 Perfect square trinomials A 2 AB B A B 2 2 2 College Algebra R.4 Factoring Polynomials Factoring Special Forms College Algebra R.4 Factoring Polynomials Factoring Special Forms Sum or difference of two cubes 2 2 A B A B A AB B 3 3 A B A B A AB B 3 3 2 2 College Algebra R.4 Factoring Polynomials Factoring Special Forms Sum or difference of two cubes 8x 125 3 27r 64 3 College Algebra R.4 Factoring Polynomials U-substitution or placeholder substitution x 2 2 2x 2 x 2x 3 2 College Algebra R.4 Factoring Polynomials Factoring flow chart on page 41 Factoring Polynomials GCF Number of Terms Two Three Difference of Squares Trinomials (a=1) Difference of Cubes Sum of Cubes Trinomials (a=1) Four Grouping Advanced Methods (4.2) College Algebra R.4 Factoring Polynomials Classwork 6 3x 2 2x 4 n 10n 9 m 8m 5 7r 3r 3 3b 7b 9 u v2 x 3 y x 3 y a 6x c 2d 8a 3x3c2 k 2 2 5 y xyz3 9x 2 z 2z 3 9 College Algebra R.4 Factoring Polynomials Homework pg 41 (1-60) College Algebra R.5 Rational Expressions FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS If P, Q, and R are polynomials, where Q or R = 0 then, PR P QR Q P PR and Q QR 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 P R PR Q S QS 1. Factor all numerators and denominators 2. Reduce common factors 3. Multiply (numerator x numerator) (denominator x denominator) College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS 1. 2. 3. Factor all numerators and denominators Reduce common factors Multiply (numerator x numerator) (denominator x denominator) College Algebra R.5 Rational Expressions MULTIPLYING RATAIONAL EXPRESSIONS 1. 2. 3. Factor all numerators and denominators Reduce common factors Multiply (numerator x numerator) (denominator x denominator) College Algebra R.5 Rational Expressions DIVISION OF RATAIONAL EXPRESSIONS P R P S PS Q S Q R QR 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 OF RATAIONAL EXPRESSIONS 1. 2. 3. 4. Find LCD of all denominators Build equivalent expressions Add or subtract numerators Write the result in lowest terms College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS College Algebra R.5 Rational Expressions ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS College Algebra R.5 Rational Expressions Simplifying Compound Fractions 2 3 3m 2 ? 3 1 2 4m 3m College Algebra R.5 Rational Expressions Homework pg 50 (1-84) College Algebra R.6 Radicals and Rational Exponents The square root of 2 2 a, a a a : a For any real number The cube root of a : a 3 For any real number 3 a, 2 3 3 a a 3 College Algebra R.6 Radicals and Rational Exponents The nth root of n n a : a n For any real number a, n an a n a a when a is odd n when a is even College Algebra R.6 Radicals and Rational Exponents RATIONAL EXPONENTS If n R is a real number with n Z and n 2 , then n 3 3 8w ? 27 R n R1 R 1 n College Algebra R.6 Radicals and Rational Exponents RATIONAL EXPONENTS For m, n Z with m and n relatively prime and n 2 m n R m n n R m n R R m College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Product property of radicals n AB A B and n n n A B AB n n College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Product property of radicals College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Quotient property of radicals n 3 A n A n B B 81 ? 3 125x n and n A n A B B College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions Rationalizing the denominator College Algebra R.6 Radicals and Rational Exponents PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS Operations on Radical Expressions Rationalizing the denominator College Algebra R.6 Radicals and Rational Exponents Homework pg 64 (1-62) College Algebra R Review State true or false. HR True 2 Q False Simplify by combining like terms 4 x x 2 x x3 x 2 6x x 2 College Algebra R Review Simplify 2a a b 3 2 2 4 3 4a12b12 1 8 x z 25x 2 60x 27 4a 6a a 9 4 2 College Algebra R Review Simplify 3x3 13x 2 x 6 m3 3mm 2 b2 2 9b College Algebra R Review Simplify r 6 4r 2 r 10 12 6 8 p 4 10p 75 College Algebra R Review Factor 9m3q 3q 4 4q 2 2 p 2 62n 9n 3 7 x 2 3 4 x 5 College Algebra R Homework pg 68 1-20