* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 1
Survey
Document related concepts
History of mathematical notation wikipedia , lookup
Bra–ket notation wikipedia , lookup
Classical Hamiltonian quaternions wikipedia , lookup
Big O notation wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Positional notation wikipedia , lookup
Law of large numbers wikipedia , lookup
Surreal number wikipedia , lookup
Factorization wikipedia , lookup
Hyperreal number wikipedia , lookup
Large numbers wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Location arithmetic wikipedia , lookup
Real number wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Division by zero wikipedia , lookup
Transcript
Chapter 1 The Real Number System Section 1.3 – Real Numbers and the Number Line Homework Problems: 7-9, 12, 15, 16, 23-25, 28-29, 34-36, 46-48 Additional Problems 1) Indicate by letter which sets the following numbers to: 5 a) Real numbers b) Irrational Numbers 1 7 c) Rational Numbers 2 d) Integers e) Whole numbers 19 f) Natural numbers -13 0 6.34 99 2) Write as a positive or negative number: a) You have one hundred dollars. b) You owe one hundred dollars. 3) Put 7, 3.4, and 2 on the number line below 4) Fill the blanks below with either < (less than) or > (greater than): a) 12 ___ 14 b) 12 ___ 14 c) 3 3 4 ___ 3 9 7 d) 77 ___ 80 Page 1 of 20 Chapter 1 The Real Number System Section 1.4 & 1.5 Adding / Subtracting Real Numbers Section 1.4 homework: 14-16, 23-25, 30-32, 63, 66 Section 1.5 homework: 1, 3, 11-13, 22-24, 31-33, 58, 63 Additional Problems 1) Add (+3) + (+4) = 2) Add (+3) + (-4) = 3) Add (-3) + (+4) = 4) Add (-3) + (-4) = 5) Subtract (+3) - (+4) = 6) Subtract (+3) - (-4) = 7) Subtract (-3) - (+4) = 8) Subtract (-3) - (-4) = 9) What is 3 + 4 + (-5) - (-2) = Page 2 of 20 Chapter 1 The Real Number System Section 1.6 Multiplying and Dividing Real Numbers Homework: 9, 12-16, 23-27, 31, 33-35, 48-50 Additional Problems 1) Find the product (reduce fractions): a) 8(7) = b) 4(3) = c) 12(0) = d) 12.2(3.2) = 4 7 e) − 3 (2) = 2) Find the quotient (reduce fractions): 42 a) 7 = b) c) −42 = 12 723.25 −1.1 = 3) Evaluate (reduce fractions): 4−14 a) 2(3)−1 = b) c) 3(11)−48 10 4(8)−103 10−3(7) = = 4) For a = 1, b = 2, c = 3 evaluate (reduce fractions) 2𝑎−𝑏 a) 3𝑐−𝑎𝑏 = b) c) 𝑏−3𝑐 𝑏𝑐−4𝑎 = 𝑎𝑐−𝑏(𝑏) 𝑐(𝑐)+𝑎𝑏 = Page 3 of 20 Chapter 1 The Real Number System Section 1.2 Variables, Expressions, and Equations Homework: 1-3, 9, 11-13, 23, 47-50, 67-70 Additional Problems 1) Evaluate the following expressions when x = 3, y = 5, and z = 6: a) 4x – 7 b) 7x – 3y c) (4z – 2y – x)(z – 1) 3𝑎−2𝑏 2) Evaluate the expression 𝑎+𝑏+1 when: a) a = 4 and b = 1 b) a = 2 and b = 6 3) For each of the following equations, is x = 6 a solution? a) 7x = 43 b) 3x – 4 = 14 c) 48 – 3x = 42 Page 4 of 20 Chapter 1 The Real Number System Section 1.7 Properties of Real Numbers Homework: 1-8, 31, 33, 35, 37, 43, 46, 49 Additional Problems 1) Rewrite 1 + (2 + 3) using a) The commutative property: b) The associative property 2) For 5, what is the a) The additive inverse? b) The multiplicative inverse? 3) What is the identity for a) Addition? b) Multiplication? 4) Rewrite using the commutative property: a) 3 + 5 = b) 3 × 5 = 5) Rewrite using the associative property: a) 3 + (5 + 7) = b) (3 × 5) × 7 = 6) Use the distributive property to expand –4(5 – x) = Page 5 of 20 Chapter 1 The Real Number System Page 6 of 20 Section 1.1 Exponents, Order of Operations, and Inequality Homework: 7, 11, 15, 25, 33, 40, 48, 51, 53, 59, 62, 77-80 Additional Problems 1) Evaluate: a) 24 = b) 53 = c) 92 = 2) Evaluate: a) (4 + 3)(5 − 2) = b) 6 ÷ 2 + 1 = c) 2+32 9−22 = 3) List which comparison symbols (from choice of =, <, >, ≠, ≤, ≥) that would form a true statement if placed between the number below (in the blank): a) 10 ___ 12 b) 11.12 ___ 11.021 c) 4 __ 4 Chapter 1 The Real Number System Section 1.8 Simplifying Expressions Homework: 2, 3, 8-10, 15-17, 27-30, 41-43 Additional Problems 1) Identify the following pair of term as like or unlike: a) 4x, 7x b) 4𝑥, −7𝑥 2 . c) 4x, 7y d) 4mar, 7ram 2) Simplify each expression (combine like terms) a) 4r – 2r = b) 4r – 2s – 5s +7r = 3) Give the numerical coefficient of each term: a) 4r b) –2s c) –5s d) 7r 4) Simplify each expression (expand and combine like terms) a) 4(𝑎2 − 3𝑎) + 2𝑎 − 1 = b) 3(𝑥 − 𝑦) − 2(𝑦 − 2𝑥) = c) 4𝑎 + 𝑏 − 3(𝑎 + 𝑏) = Page 7 of 20 Chapter 1 The Real Number System Page 8 of 20 Vocabulary 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) absolute (value) add additive (inverse) algebraic associative base coefficient combine commutative comparison denominator difference distributive division divisor equality equations evaluate exponents expression factors greater (than) grouping identify identity integers irrational less (than) like (terms) lowest minuend multiplication (property) 33) multiplicative (inverse) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) multiplying natural (numbers) negative number (line) numerator numerical operations opposite order ordering power product property quotient raised rational reciprocal set set-builder (notation) signed simplifying solutions subtract subtrahend sum symbol terms undefined unlike variable whole (numbers) Chapter 1 The Real Number System Section 1.3 – Real Numbers and the Number Line Additional Answers 1) Indicate by letter which sets the following numbers to: 5 5 a, c, d, e, f a) Real numbers 1 b) Irrational Numbers 1 7 a, c 7 c) Rational Numbers 2 2 d) Integers a, b √19 e) Whole numbers 19 f) Natural numbers -13 -13 a, c, d 0 0 a, c, d, e 6.34 6.34 a, c 99 99 a, c. d, e, f 2) Write as a positive or negative number: c) You have one hundred dollars. 100 d) You owe one hundred dollars. -100 3) Put 7, 3.4, and 2 on the number line below 4) Fill the blanks below with either < (less than) or > (greater than): e) 12 _<_ 14 f) -12 _>_ -14 3 4 g) 3 7 _<_ 3 9 h) 77 _>_ -80 Vocabulary: absolute (value) additive (inverse) greater (than) integers irrational less (than) natural (numbers) negative number (line) opposite ordering rational set set-builder (notation) whole (numbers) Page 9 of 20 Chapter 1 The Real Number System Section 1.4 & 1.5 Adding / Subtracting Real Numbers Additional Answers 1) 2) 3) 4) 5) 6) 7) 8) 9) Add (+3) + (+4) = 7 Add (+3) + (4) = 1 Add (3) + (+4) = 1 Add (-3) + (-4) = Subtract (+3) (+4) = Subtract (+3) - (-4) = 7 Subtract (-3) - (+4) = Subtract (-3) - (-4) = 1 What is 3 + 4 + (-5) - (-2) = 7 3 = 4 Vocabulary: add difference minuend signed subtract subtrahend sum Page 10 of 20 Chapter 1 The Real Number System Section 1.6 Multiplying and Dividing Real Numbers Additional Answers 1) Find the product (reduce fractions): f) 8(7) = 56 g) 4(3) = 12 h) 12(0) = 0 i) 12.2(3.2) = 39.04 4 7 28 𝟏𝟒 j) − 3 (2) = − 6 = − 𝟑 2) Find the quotient (reduce fractions): 42 d) 7 = 𝟔 e) f) −42 𝟕 = −𝟐 12 723.25 −1.1 = −𝟔. 𝟓𝟕𝟓 = − 𝟐𝟔𝟑 𝟒𝟎 , Note: Either answer is OK. 3) Evaluate (reduce fractions): 4−14 −10 d) 2(3)−1 = 5 = −𝟐 e) f) 3(11)−48 10 4(8)−103 10−3(7) = −15 10 −71 𝟑 = − 𝟐 = −𝟏. 𝟓, Note: Either answer is OK. 𝟕𝟏 = −11 = 𝟏𝟏 3) For a = 1, b = 2, c = 3 evaluate (reduce fractions) d) e) f) 2𝑎−𝑏 2(1)−(2) 2−2 0 = 3(3)−(1)(2) = 9−2 = 7 = 𝟎 3𝑐−𝑎𝑏 𝑏−3𝑐 (2)−3(3) 2−9 −7 𝟕 = (2)(3)−4(1) = 6−4 = 2 = − 𝟐 𝑏𝑐−4𝑎 (1)(3)−(2)(2) 𝑎𝑐−𝑏(𝑏) 3−4 −𝟏 𝑐(𝑐)+𝑎𝑏 = 3(3)+(1)(2) = 9+2 = Vocabulary: denominator division divisor lowest multiplication (property) multiplying numerator product quotient reciprocal terms undefined 𝟏𝟏 = −𝟑. 𝟓, Note: Either answer is OK. 𝟏 = − 𝟏𝟏, Note: Either answer is OK. Page 11 of 20 Chapter 1 The Real Number System Page 12 of 20 Section 1.2 Variables, Expressions, and Equations Additional Problems 1) Evaluate the following expressions when x = 3, y = 5, and z = 6: d) 4x – 7 4(3) – 7 = 12 – 7 = 5 e) 7x – 3y 7(3) – 3(5) = 21 – 15 = 6 f) (4z – 2y – x)(z – 1) [4(6) – 2(5) – (3)][(6) – 1] = (24 – 10 – 3)(5) = 11(5) = 55 3𝑎−2𝑏 2) Evaluate the expression 𝑎+𝑏+1 when: c) a = 4 and b = 1 d) a = 2 and b = 6 3(4)−2(1) (4)+(1)+1 3(2)−2(6) (2)+(6)+1 = = 12−2 6 6−12 9 = = 𝟏𝟎 𝟔 −𝟔 𝟗 𝟓 =𝟑 =− −𝟐 𝟑 Note: Do not have to reduce unless instructions ask you to reduce answers. 3) For each of the following equations, is x = 6 a solution? d) 7x = 43 7(6) = 43 42 = 43 e) 3x – 4 = 14 3(6) – 4 = 14 18 – 4 = 14 14 = 14 f) 48 – 3x = 42 48 – 3(6) = 42 48 – 18 = 42 30 = 42 Vocabulary: algebraic equality equations evaluate expression identify solutions symbol variable No Yes No Chapter 1 The Real Number System Section 1.7 Properties of Real Numbers Additional Answers 1) Rewrite 1 + (2 + 3) using c) The commutative property: d) The associative property 2) For 5, what is the c) The additive inverse? d) The multiplicative inverse? 3) What is the identity for c) Addition? d) Multiplication? = (2 + 3) + 1 or = 1 + (3 + 2) = (1 + 2) + 3 5 𝟏 𝟓 0 1 4) Rewrite using the commutative property: c) 3 + 5 = 5 + 3 d) 3 × 5 = 5 × 3 5) Rewrite using the associative property: c) 3 + (5 + 7) = (3 + 5) + 7 d) (3 × 5) × 7 = 3 × (5 × 7) 6) Use the distributive property to expand –4(5 – x) = –20 + 4x Vocabulary: associative commutative distributive identity multiplicative (inverse) property Page 13 of 20 Chapter 1 The Real Number System Page 14 of 20 Section 1.1 Exponents, Order of Operations, and Inequality Additional Problems 1) Evaluate: d) 24 = 16 e) 53 = 125 f) 92 = 81 2) Evaluate: d) (4 + 3)(5 − 2) = (7)(3) = 𝟐𝟏 e) 6 ÷ 2 + 1 = 3 + 1 = 𝟒 f) 2+32 9−22 = 2+9 9−4 = 𝟏𝟏 𝟓 3) List which comparison symbols (from choice of =, <, >, ≠, ≤, ≥) that would form a true statement if placed between the number below (in the blank): d) 10 ___ 12 <, ≠, ≤ e) 11.12 ___ 11.021 >, ≠, ≥ f) 4 __ 4 =, ≤, ≥ Vocabulary: base comparison exponents grouping operations order power raised Chapter 1 The Real Number System Section 1.8 Simplifying Expressions Homework: 7, 11, 15, 25, 33, 40, 48, 51, 53, 59, 62, 77-80 Additional Problems 1) Identify the following pair of term as like or unlike: e) 4x, 7x like 2 f) 4𝑥, −7𝑥 unlike g) 4x, 7y unlike h) 4mar, 7ram like (put in alphabetical order and moth are arm) 2) Simplify each expression (combine like terms) c) 4r – 2r = 2r d) 4r – 2s – 5s +7r = 11r – 7s 3) Give the numerical coefficient of each term: e) 4r 4 f) –2s –2 g) –5s –5 h) 7r 7 4) Simplify each expression (expand and combine like terms) d) 4(𝑎2 − 3𝑎) + 2𝑎 − 1 = 4𝑎2 − 12𝑎 + 2𝑎 − 1 = 𝟒𝒂𝟐 − 𝟏𝟎𝒂 − 𝟏 e) 3(𝑥 − 𝑦) − 2(𝑦 − 2𝑥) = 3𝑥 − 3𝑦 − 2𝑦 + 4𝑥 = 𝟕𝒙 − 𝟓𝒚 f) 4𝑎 + 𝑏 − 3(𝑎 + 𝑏) = 4𝑎 + 𝑏 − 3𝑎 − 3𝑏 = 𝒂 − 𝟐𝒃 Vocabulary: coefficient combine factors like (terms) numerical simplifying terms unlike Page 15 of 20 Chapter 1 The Real Number System Page 16 of 20 Glossary Absolute value: (See number) Addends: (See Addition). Addition: is combining several quantities together. Note: For real numbers, addition can be considered as combining lengths together. From where you are, move the length (size of the number) you want to go in the direction of the number. At the beginning you start from zero. Addends: are the quantities being added together. Sum: the result of addition. Associative law of addition: says that the order that the results of combining the addends together is independent of the order you add the addends together. (Just make sure you use each addend once and only once). Commutative property of addition: says that adding the first number to the second produces the same result as adding the second number to the first. Zero Property of addition: states that adding zero to any number results in the same number. For the reason, zero is called the additive identity. Additive Identity: Zero. Adding zero does not change the result. Algebraic Expression: is a finite collection of numbers, variables, groupings combined together with the operations of addition, subtraction, multiplication, division, and exponentiation. Note: This can be extended to include algebraic functions (which can be expressed as an algebraic expression in just the variables of the function). Normally the exponents are rational, because otherwise one tends to get outside the algebraic realm. Associative Property: is when, for the same operation, grouping (what you do first) does not make any difference. (See Addition and Multiplication). Base: (See Exponential Notation) Coefficient: (See Terms) Combine like terms: (See terms) Commutative property: is when for a numerical operation the numbers can be reversed and the result is the same. I.E., A operation B has the same value as B operation A. (See Addition and multiplication). Comparison Relations: Give an indication which of a pair of numbers is larger and or indicate they could be the same value. We say a number is larger than another number if there is a positive number that can be added to the second number to give the first number. Equal (=): Says the two values on either side of the equal relation have the same value. Greater than (>): Says the value on the left of the greater than relation is larger than the value on the right. Greater than or equal to (≥): Says the value on the left of the greater than or equal to relation is either larger than or equal to the value on the right. Less than (<): Says the value on the right of the greater than relation is larger than the value on the left. Less than or equal to (≤): Says the value on the right of the greater than or equal to relation is either larger than or equal to the value on the left. Not equal to (≠): Says the two values on either side of the not equal relation have different values. There is no indication which is larger. Denominator: (See Fraction) Difference: (See Subtraction) Distributive law of multiplication: (See Multiplication) Dividend: (See Division) Chapter 1 The Real Number System Page 17 of 20 Division: is the inverse operation of multiplication. To divide by a number, multiply by its reciprocal. Dividend: is the number being divided. In the expression 30 divided by 10 (or 10 into 30), 30 is the dividend. Divisor: is the number doing the dividing. In the expression 30 divided by 10 (or 10 into 30), 10 is the divisor. Reciprocal: To find the reciprocal of a value, write the number as a fraction and then interchange the numerator and denominator. Note: For mixed numbers, you need to rewrite 1 the number as an improper fraction first. The reciprocal of n is 𝑛. Quotient: is the (principal) result of division. In the expression 30 divided by 10 is 3, 3 is the quotient. This class we will not be using remainders. Note: Division by zero is undefined. Note: If you divide zero by any number, except zero, the quotient is zero. Divisor: (See Division) Element: (See Set) Equal (=): (See Comparison Relations) Equation: Is a mathematical statement that states two expressions have the same value. Solutions of an equation: is the set of points (values for the variables) that make the equation a true statement. Note: In a manner of speaking, an equation is something we would like to be true. Finding (hopefully all) the solutions of the equation is called solving the equation. Evaluate: 1) Simplify an expression to determine its value. 2) Substitute given values for variables in an mathematical expression, and the determine 2+𝑥 the value for that expression. Example: To evaluate 4+𝑥, when x = 6. First replace x in 2+(6) 8 4 that expression by (6) to get 4+(6). Next evaluate that expresion to get 10 = 5. Exponent: (See Exponential Notation and Order of Operations) Exponential Notation: Indicates that a number is to be multiplied by itself several times. Base: is the number that is to be multiplied by itself several times. Exponent: Indicates how many times a number is to be multiplied by itself. Example: The notation 53 indicates that 5, the base, is to be multiplied by itself 3 (the exponent) times; i.e. 53 5 5 5 25 5 125 . Power: is a synonym for exponent. Normally the word power is preceded by an ordinal number. 54 is read as “five to the fourth power”. For the exponent of 2, the number can be read as “squared” instead of “to the second power”. For the exponent of 3, the number can be read as “cubed” instead of “to the third power”. Factors: (See multiplication). Fraction: In this course, this is the usual way of doing division. It is normally seen as 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 . 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 Denominator: is the bottom half of the fraction, it is the divisor of the division operation. Fraction bar: Is the line that separates the numerator from the denominator. It is also a grouping symbol (see Order of Operations). Numerator: is the top half of the fraction, it is the dividend of the division operation. Greater than (>): (See Comparison Relations) Greater than or equal to (≥): (See Comparison Relations) Groupings: (See Order of Operations) Chapter 1 The Real Number System Page 18 of 20 Identity: (See Additive Identity and Multiplicative Identity). Also refers to an expression that is always true as x + x = 2x. Integers: Any number that can be rewritten (is equal to a number) as having no fractional part or digits past the decimal point. Natural number: is a positive integer – i.e. integers starting with one. They are sometimes referred to as the counting numbers. Whole number: is a nonnegative integer – i.e. integers starting with zero. Irrational: (See Rational Number) Less than (<): (See Comparison Relations) Less than or equal to (≤): (See Comparison Relations) Like terms: (See Terms) Minuend: (See Subtraction) Multiplicand: (See Multiplication) Multiplicative Identity: is one. Anything multiplied (or divided by 1) does not change its value. Multiplication: In the expression 2 times 12 is 24: Multiplicand: 2 is the multiplicand, the number being multiplied. Multiplier: 12 is the multiplier; multiply the multiplicand by the multiplier. Product: 24 is the product, the result of multiplication. Factors: are things multiplied together to produce a product. Both the multiplicand and the multiplier are considered a factor of the product. Distributive law of multiplication: says that adding (or subtracting) two numbers together then multiply the result by a third number produces the same result as multiplying the first two numbers by the third number first then adding (or subtracting) the two products together. Example: 6 (2 3) 6 5 30 and (6 2) (6 3) 12 18 30 . Zero property of multiplication: refers to the fact that the product is zero if any of the factors of the product are zero. Multiplier: (See Multiplication) Natural number: (See Integers) Negative: (See Number) Not equal to (≠): (See Comparison Relations) Number: A number has a size and direction. Absolute value: indicates the size of the number. When the number represents length, the absolute value of a number indicates how far the number is from the value zero. Sign: indicates the direction of the number. A positive sign (+) means it is in the direction of numbers getting larger – on the number line that is normally to the right. A negative sign (–) means it is in the direction of numbers getting smaller – on the number line that is normally to the left. Zero has no direction associated with it. However, one may write zero as either 0, +0, or –0. Note: For a non-zero number, if the sign is omitted, the sign is assumed to be positive. Number line: is used to represent numbers as points on a straight line. The distance between two numbers is proportional to the difference between the two numbers. On the standard number line, the line is horizontal with larger numbers on the right. For a vertical number line, the larger numbers are higher and the smaller numbers are lower. Note: The number line can at best represent just part of the real numbers. Zero does not have to be on any number line you draw. You are also free to choose a desirable scale. Numerator: (See Fraction) Numerical Coefficient: (See Terms) Chapter 1 The Real Number System Page 19 of 20 Order of operations: dictates the order an arithmetic expression is correctly evaluated. In brief the order is: Groupings as parenthesis ( ), brackets [ ], or braces {}. Essentially this means to work from the inside to the outside. Other examples of (implied) groupings are denominators of fractions, numerators of fractions, expressions under the radical / square root sign, 3 5 expressions in the exponent (as 2 , you evaluate the exponent 3+5 first). In fact anything used as a subscript, superscript, or index, or parameter needs to be evaluated before it is used. Exponents, evaluating a base raised to a power is done next; Multiplication and Division is done next. Either of these two may be done first. (However, for ease of calculation, we suggest doing multiplications before divisions. Note you can rearrange the order of multiplying and dividing so the divisions come last); Addition and Subtraction are done last. Note: Because the addition and subtractions are done last, you can evaluate everything between the outermost addition and subtraction signs separately from each other and you will still get the correct evaluation. Note there are other rules, like the distribution law of multiplication, which allows one to restate the expression before evaluating it. Positive: (See number) Power: (See Exponential Notation) Product: (See Multiplication) Quotient: (See Division) Rational Number: is any number that can be written as the quotient of two integers. If a number cannot be written as the quotient of two integers, it is called irrational. Reciprocal: (See Division) Set: is a collection of objects. Element: is what each objects in the set is called. Set-list notation: The name of each element in the set is listed inside braces as {1, 3, 5, 7}. Set-Builder notation: describes the members in the set instead of listing the members. Example: {n | n is an even whole number}. The prior example is read as “The set n where n is an even whole number”. Sign: (See number) Solutions of an equation: (See Equation) Subtraction: adds the opposite to a quantity. Note: For real numbers, subtraction can be considered as combining lengths together. From where you are, move the length (size of the number) you want to go in the opposite direction of the number. At the beginning you start from zero. Difference: the result of subtraction. In the expression 5 – 3 = 2, 2 is the difference. Sometimes the difference is called the remainder. Minuend: In the expression 5 – 3 = 2, 5 is the minuend. Subtrahend: In the expression 5 – 3 = 2, 3 is the subtrahend. Subtrahend: (See Subtraction) Sum: (See Addition) Symbol: (See variable) Chapter 1 The Real Number System Page 20 of 20 Terms: Are things (numbers, variables) that are algebraically added or subtracted together. Note: there is some ambiguity in this definition. 5xy is an algebraic expression with just one term; it is not added or subtracted to anything, but could be. That term contains three factors: 5, x, and y. 4x + 5y – 2xy contain three terms: 4x, 5y, and 2xy. The easiest way to count the number of terms is to count the outermost addition (+) and subtraction (–) operators and add 1 to the sum. Outermost here means not inside a grouping symbol. The confusion starts when you find out that for 2(x + 2y + 3z) = 2x + 4y + 6z , the expression on the left contains just one term and the expression on the right contains 3 terms. But the right side of the equation is just the expansion of the left hand side! By the way the factor (x + 2y + 3z) contains 3 terms inside parenthesis. Numerical Coefficient: In a term, there is normally just one outermost numerical factor. That number is called the numerical coefficient. For the term 5xy, the numerical coefficient is 5. If there is no visible number, then the numerical coefficient is considered to be 1 since ab = 1ab. We will avoid weird expressions as 2x5y in this course. (Not sure if the numerical coefficient should be 2, 5, or 10). However, for the term 2x(5y) the numerical coefficient is 2 since 5 is not “outermost”. The numerical coefficient is often just called the coefficient. Like terms: Two terms in an algebraic expression are considered like terms if their nonnumerical factors can be rearranged to be the same. Remember an exponent indicates that a factor has been repeated that many times. Since multiplication is commutative, it is easiest to list the factors in alphabetical order (assuming alphabetical variables) and check if they are the same. Example 2mar would become 2amr, – 3ram,would become – 3amr, and 7arm would become 7amr; so all three of those terms are like terms. However, the term 5𝑎2 𝑚𝑟 woulb become 5aamr is not like the other three terms because it has an extra factor of a. Combine like terms: Like terms can be added or subtracted together by just adding subtracting the numerical coefficients together and then placing the common factors after the sum. Example: 3x + 4y + 5 – 6y + 2x = 5x – 2y + 5 since for the like terms 3x + 2x we have 3 + 2 = 5 and for the like terms + 4y – 6y we have + 4 – 6 = –2. Unlike terms: If two different terms are not like terms, they are called unlike terms. Undefined: (See Division) Unlike terms: (See terms). Variable: is a symbol, usually a letter of (any) alphabet that represents (or can be replaced by) a number. Whole number: (See Integers) Zero property of multiplication: (See Multiplication).