Mathematics Tutorials These pages are intended to aide in the preparation for the Mathematics Placement test. They are not intended to be a substitute for any mathematics course. Arithmetic Tutorials Algebra I Tutorials Algebra II Tutorials Word Problems
Arithmetic Tutorials Whole Numbers Sets of Numbers Properties of Real Numbers Addition of Whole Numbers Subtraction of Whole Numbers Multiplication of Whole Numbers Division of Whole Numbers Order of Operations Exponential Notation Prime Numbers and Factoring Roman Numerals Decimals Introduction to Decimals Addition of Decimals Subtraction of Decimals Multiplication of Decimals Division of Decimals Converting Fractions to Decimals Scientific Notation Percents Introduction to Percents Solving Percent Equations Fractions Least Common Multiple Greatest Common Factors Reducing Fractions Addition of Fractions and Mixed Numbers Subtraction of Fractions and Mixed Numbers Multiplication of Fractions and Mixed Numbers Division of Fractions and Mixed Numbers Converting between Mixed Numbers and Improper Fractions Ratio and Proportions Introduction to Ratios Introduction to Rates Introduction to Proportions Signed Numbers Negative Numbers Addition and Subtraction of Signed Numbers Multiplication and Division of Signed Numbers Return to Main Page
Algebra I Tutorials Linear Equations and Inequalities What is a Variable? Evaluating Algebraic Expressions Combining Like Terms in Algebraic Expressions Properties of Equalities Algebraic Equations Solving First Degree(Linear) Equations Literal Equations and Formulas Linear Inequalities Exponents Multiplication with Exponents Division with Exponents Polynomials What is a Polynomial? Evaluating a Polynomial Addition of Polynomials Subtraction of Polynomials Multiplication of Polynomials Special Products Division of Polynomials Factoring Polynomials Common Factors Factoring using Common Factors Factoring by Grouping Solving Equations by Factoring Factoring the difference of two squares Factoring the sum or difference of cubes Return to Main Page
Algebra II Tutorials Rational Expressions Introduction to Rational Expressions Equivalent Rational Expressions Simplifying Rational Expressions Addition and Subtraction of Rational Expressions Multiplication and Division of Rational Expressions Complex Fractions Equations involving Fractions Linear Equations Graphing Linear Equations Cartesian Coordinate System Graphs of Linear Equations Intercepts Slope of the Line Systems of Linear Equations Introduction to Systems of Linear Equations Solving by Substitution Solving by Elimination Quadratic Equations Introduction to Quadratic Equations Solving Quadratic Equations Factoring Extracting the Root Completing the Square Quadratic Formula Complex Numbers Introduction to Complex Numbers Addition and Subtraction of Complex Numbers Multiplication of Complex Numbers Division of Complex Numbers Complex Solutions to Quadratic Equations Roots and Radicals Common Roots and Radicals Properties of Radicals Addition and Subtraction of Radicals Multiplication of Radicals Division of Radicals Equations involving Radicals Return to Main Page
Word Problems Translating Word Problems to Algebra Number Problems Age Problems Coin Problems Work Problems Mixture Problems Distance Problems Return to Main Page
Mathematical Numbers Natural Numbers Natural numbers, also known as counting numbers, are the numbers beginning with 1, with each successive number greater than its predecessor by 1. If the set of natural numbers is denoted by N, then N = { 1, 2, 3,...} Whole Numbers Whole numbers are the numbers beginning with 0, with each successive number greater than its predecessor by 1. It combines the set of natural numbers and the number 0. If the set of whole numbers is denoted by W, then W= { 0, 1, 2, 3,...} Rational and Irrational Numbers Rational numbers are the numbers that can be represented as the quotient of two integers p and q, where q is not equal to zero. If the set of rational numbers is denoted by Q, then Q = { all x, where x = p / q, p and q are integers, q is not zero} Rational numbers can be represented as: (1) Integers: (4 / 2) = 2, (12 / 4) = 3 (2) Fractions: 3 / 4, 13 / 3 (3) Terminating Decimals: (3 / 4) = 0.75, (6 / 5) = 1.2 (4) Repeating Decimals: (13 / 3) = 4.333... (4 / 11) =.363636... Conversely, irrational numbers are the numbers that cannot be represented as the quotient of two integers, i.e., irrational numbers cannot be rational numbers and vice-versa. If the set of irrational numbers is denoted by H, then H = { all x, where there exists no integers p and q such that x = p / q, q is not zero } Typical examples of irrational numbers are the numbers π and e, as well as the principal roots of rational numbers. They can be expressed as non-repeating decimals, i.e., the numbers after the decimal point do not repeat their pattern. Real Numbers Real numbers are the numbers that are either rational or irrational, i.e., the set of real numbers is the union of the sets Q and H. If the set of real numbers is denoted by R, then
R = Q H Since Q and H are mutually exclusive sets, any member of R is also a member of only one of the sets Q and H. Therefore; a real number is either rational or irrational (but not both). If a real number is rational, it can be expressed as an integer, as the quotient of two integers, and a terminating or repeating decimal can represent it; otherwise, it is irrational and cannot be represented in the above formats. Complex Numbers Complex numbers are the numbers with the format a + b i, where a and b are real numbers and i² = - 1. If we denote the set of complex numbers by C, then C = { a + b i, where a and b are real numbers, i² = -1 } If in the number x = a + b i, b is set to zero, then x = a, where a is a real number. Thus, all real numbers are complex numbers, i.e., the set of complex numbers includes the set of real numbers.
Real Number System The real number system is comprised of the set of real numbers and the arithmetic operations of addition and multiplication (subtraction, division and other operations are derived from these two). The rules and relationships that govern the real number system are the basis for most algebraic manipulations. Properties of Real Numbers All real numbers have the following properties: (1) Reflexive Property For any real number a, a = a. Example: 3 = 3, y = y, x + z = x + z (x, y and z are real numbers) (2) Symmetric Property For any real numbers a and b, if a = b, then b = a. Example: If 3 = 1 + 2, then 1 + 2 = 3 (3) Transitive Property For any real numbers a, b and c, if a = b and b = c, then a = c. Example: If 2 + 3 = 5 and 5 = 1 + 4, then 2 + 3 = 1 + 4. (4) Substitution Property For any real numbers a and b, if a = b, then a may be replaced by b, and b may be replaced by a, in any mathematical statement without changing the meaning of the statement. Example: If a = 3 and a + b = 5, then 3 + b = 5. (5) Trichotomy Property For any real numbers a and b, one and only one of the following conditions holds: (1) a is greater than b ( a > b) (2) a is equal to b ( a = b) (3) a is less than b ( a < b) Example: 3 < 4, 4 + 2 = 6, 7 > 5 Absolute Values The absolute value of a real number is the distance between its corresponding point on the number line and the number 0. The absolute value of the real number a is denoted by a.
From the diagram, it is clear that the absolute value of nonnegative numbers is the number itself, while the absolute value of negative integers is the negative of the number. Thus, the absolute value of a real number can be defined as follows: For all real numbers a, (1) If a >= 0, then a = a. (2) If a < 0, then a = -a. Examples: 2 = 2-4.5 = 4.5 0 = 0
Addition of Whole Numbers Addition is the process of finding the total of two or more numbers. We first learn addition through counting *** + **** = ******* 3 4 7 We can also look at addition on our old friend the number line. each of these line segments is 3 units long (length does not depend upon position 7(total) 3 + 4 Therefore, we also have 3 + 4 = 7 There a few special properties of addition that we need to be aware of. Addition Property of Zero Zero added to any number does not change the number 4 + 0 = 4 0 + 1 = 1 Commutative Property of Addition Two numbers can be added in any order, the sum is the same 4 + 8 = 12 8 + 4 = 12 Associative Property of Addition Grouping of addition in any order does not change the sum (4 + 2) + 3 = 6 + 3 = 9 4 + (2 + 3) = 4 + 5 = 9 The number line and other assorted aides are fine to learn with but the basic addition of 2 one-digit numbers must be memorized. The addition of larger numbers is basically the repeated usage of the basic addition facts.
Addition Table + 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 To use this table to add two numbers, find the first number to be added in the top row (put your finger there). Find the second number to be added in the first column (put another finger there). Now, bring your fingers together (first finger straight down and second finger straight across). The place where your fingers meet is the sum. Addition of larger numbers The easiest way to add larger numbers is to arrange the numbers vertically, keeping the digits of the same place value in the same column. Add 321 + 6472 321 +6472 6793 Addition with carry If one of the columns in an addition sums to a value greater than 9 then you must perform a carry. Write down the one s digit of the sum and carry the ten s digit into the next column to the left. Add 645 + 476 645 +476 5 + 6 = 11 1 645 +476 1 + 4 + 7 = 12 1 1 645 +476 1 + 6 + 4 = 11 1121
When do we use Addition? There are several types of problems that require the use of addition. One of the major clues to the use of addition is the major key words leading to addition. Addition Key Words Added to 3 added to 5 3 + 5 More than 7 more than 5 5 + 7 The sum of the sum of 3 and 9 3 + 9 Increased by 4 increased by 6 4 + 6 The total of the total of 3 and 8 3 + 8 Plus 5 plus 10 5 + 10
Subtraction of Whole Numbers Subtraction is the process of finding the difference between two numbers. We learn subtraction (as with addition) by counting. ******** - ******** = ***** 8 3 5 Minuend Subtrahend Difference We can also show subtraction on the familiar number line. 8 3 5 You can readily see that addition and subtraction are related Subtrahend +Difference = Minuend 3 + 5 = 8 You can use this fact to check you subtraction with addition. Subtraction of Larger Numbers To perform subtraction on larger numbers by arranging the numbers vertically ( as in addition). Then subtract the numbers in each column. Subtract 8955 2432 8955-2432 6523
Subtraction with borrowing If during the course of perform a subtraction on a large number, you are attempting to subtract a large number from a smaller number you must use borrowing. Subtract 692 378 692-378 you can not subtract 8 from 2 so we need to borrow 1 ten from the 9 tens in the tens column leaving 8 tens and 12 ones. 81 692-378 12 8 = 4 4 81 692-378 8 7 = 1 14 81 692-378 6 3 = 3 314 When do we use Subtraction? There are many key words that lead us to perform subtraction. Subtraction Key Words Minus 8 minus 5 8 5 Less 9 less 3 9 3 Less than 2 less than 7 7 2 The difference between the difference between 8 and 2 8 2 Decreased by 5 decreased by 1 5 1
Multiplying Whole Numbers Multiplication is basically repeated additions. 3 2 = 2 + 2 + 2 = 6 6 8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 48 The numbers that are multiplied are called factors (6 and 8) and the result is the product (48). There are three basic ways to represent multiplication a b a b all mean the same thing (a multiplied by b) a(b) As is addition the best way to learn multiplication is to memorize the basic facts. Multiplication Table 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 To use the table, place one finger on the top row on the first factor and place another finger on the second factor on the first column. Bring the finger together and you have the product. Properties of Multiplication There are several useful properties of multiplication that will help us in our computations Multiplication Property of Zero The product of any number and 0 is 0 0 4 = 0 7 0 = 0 Multiplication Property of One The product of any number and one is the number 1 5 = 5 6 1 = 6
Commutative Property of Multiplication Two numbers can be multiplied in either order and the product is unchanged. 4 3 = 3 4 = 12 result Associative Property of Multiplication Grouping the numbers in a multiplication problem in any order gives the same (4 2) 3 = 8 3 = 24 4 (2 3) = 4 6 = 24 Multiplying Larger Numbers Multiplying large numbers involves the repeated usage of basic one-digit multiplication facts. Multiply 37 4 3 7 4 4 7=28 write the 8 in the one s column and carry the 8 above the ten s column 2 3 7 4 8 3 4=12, add the carry digit 12 + 2 = 14 3 7 4 148 Multiply 47 23 47 23 47 3 = 141 47 23 141 47 23 141 940 1081 20 47 = 940 141 + 940 = 1081
Multiply 439 206 439 203 3 439 = 2634 439 203 2634 439 203 2634 0000 439 203 2634 00000 87800 90434 0 439 = 000 200 x 439 = 87800 2634 + 0000 + 87800 = 90434 When do we use multiplication? There are key words that indicate the use of multiplication Multiplication Key Word Times 7 times 3 7 3 The product of The product of 6 and 9 6 9 Multiplied by 8 multiplied by 2 8 2
Division of Whole Numbers Division is used to separate objects into groups of equal size. Division is the inverse of multiplication (3 4 = 12 and 12 3 = 4 and 12 4 = 3) We right division in two different ways 12 4 is the same as 4 12 6 Look at the division 4 24, we refer to 4 as the divisor, 24 as the dividend and 6 as the quotient. In general, we have quotient divisor dividend 6 Also, we can see the relationship between division and multiplication 4 24 because 4 6 = 24 Important Division Rules Any number, except zero, divided by itself equals 1 1 1 8 8 2 2 Any number divided by 1 is the number itself 8 27 1 8 1 27 Zero divided by any number is zero 0 0 7 0 101 0 Division by zero is not allowed? 0 8, there is no number whose product with 8 is 0
Dividing single digits into larger numbers 79 4 3192 28 39 36 3 4 3192 Divide 3192 4 4 divides into 31-7 times since 4 * 7 = 28 7 4 3192 subtract 28 from 31, bring down the 9 28 39 7 4 3192 28 4 divides into 39-9 times since 4*9=36, 39 subtract 39-36 36 3 7 4 3192 28 bring down the 2, 39 4 divides into 32-8 times since 4*8=32, 36 32 subtract 32 32 = 0 32 0 Dividing by single digit with a remainder 4 3 14 Divide 3 14 3*4 = 12 4 subtract 14 12 = 2 3 14 12 2 4 r 2 so the result is 4 with a remainder of 2 (3*4 3 14 + 2=14) 12 2
Divide larger numbers Divide 34 1598 34 1598 Think about 3 * 5 = 15 but 5 * 34 = 170, which is larger than 159, so use 4 4 34 1598 4*34 = 136 subtract 159 136 = 23 bring down the 8 4 34 1598 136 238 47 34 1598 136 238 238 7*34 = 238 subtract 238-238= 0 So the solution is 37 since 47 * 34 = 1598 When do we use Division? There are a couple of key words that indicate the use of division. Division Key Words The quotient of The quotient of 9 and 3 9 3 Divided by 6 divided by 2 6 2
Order of Operations Many times, in math classes, the problems involve more than one operation in the same problem. We need a system to determine the order in which we perform our operations. There is a hierarchy of operations that keep us from being confused by the messy problems. The Order of Operations can be remembered by learning the phrase Please Excuse My Dear Aunt Sally. levels) Parenthesis Exponents Multiplication & Division Addition & Subtraction (4 Ex. 4+5*6 from left to right we see addition and multiplication (multiplication first priority) 4+30 now we can add 34 Ex. 55 2*10 + 43 subtraction multiplication addition (multiplication first) 55 20 + 43 now left to right (subtraction and division equal priority) 35 + 43 78 Ex. (3 + 4) 2 parenthesis and exponent (p first) 7 2 now exponent 49
Exponential Notation Repeated multiplication of the same number can be written in two different ways 3*3*3*3 or 3 4 exponent The exponent shows how many time 3 is multiplied by itself. 3 4 is in a format called exponential notation. Examples of exponential notation 6 = 6 1 six to the first power (usually don t write the 1) 6*6 = 6 2 six squared or six to the second power 6*6*6 = 6 3 six cubed or six to the third power etc. 3*3*3*3*5*5*5 = 3 4 *5 3 Place values are actually powers of 10 Ten 10 1 Hundred 10 2 Thousand 10 3 Ten-Thousand 10 4 Hundred-Thousand 10 5 Million 10 6
Factoring Numbers We can divide whole numbers into two categories (prime and composite). Prime numbers are numbers that are only divisible by 1 and itself such as 3, 5, 11, 13. Composite numbers are numbers that are products of prime numbers such as 6, 15, 20. One of the major things that we need to do with whole numbers is to factor the composite numbers into their prime parts, called factoring. Ex. Ex. Ex. Ex. 10 = 2*5 20 = 2*2*5 Factor 105 Start with the small primes and check for divisibility 2 does not work since 2 does not divide 105 evenly but 3 works 105 = 3*35 now factor 35 as 5*7 so we get 105 = 3*5*7 Factor 129 2 won t work but 3 does 129 = 3*43, 43 is prime so 129 = 3 *43 Factor 400 400 = 2*200 FACTOR 200 400 = 2*2*100 FACTOR 100 400 = 2*2*2*50 FACTOR 50 400 = 2*2*2*5*5 DONE
Roman Numerals Prior to the development of our number system, there have been many other civilizations who have had their own unique way of handling mathematics and arithmetic. The one system that has survived to this day and is still in wide use is the Roman Numeral System. The Roman Numeral System A few major things to realize about the Roman Numeral System There is no zero It uses what we think of as letters (I, V, X, L, C, M) Placing a lower value to the left of a higher value subtracts Placing a lower value to the right of a higher value adds I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 Notice the significance of 5 in this system (like 10 in our system) See a pattern 1 = I 2 = II 3 = III 4 = IV 5 1 you add through three higher then for 4 you subtract 6 = VI 7 = VII 8 = VIII 9 = IX 10 1 LX = 60 XL = 40 50 + 10 50 10 Starting on the left you build the value MCMLXVI M = 1000 CM = 900 1000-100 LX = 60 50 + 10 VI = 6 5 + 1 1000 + 900 + 60 + 6 = 1966 you see these type of things on movie dates.
Let s go from our system to Roman 2121 2000 = MM 100 = C 20 =XX 1 = I MMCXXI
Least Common Multiple and Greatest Common Factor When working with a group of two or more numbers, we sometimes have to find two specials numbers, the least common multiple and the greatest common factor. Least Common Multiple The Least Common Multiple is the smallest number that is a multiple of each number in the group. The least common multiple of 2 and 3 is 6 since it is the smallest number both 2 and 3 divide evenly Finding the least common multiple. Factor each number Write down all the factors of the first number Add in the factors from the other numbers that are not in the LCM already Multiply all the numbers together Ex. Find LCM for 30 and 45 30 = 2*3*5 45 = 3*3*5 LCM start with 30 and write down 2*3*5 Look at 45, it has 2-3 s and a 5, the LCM has a 5 but only one 3 so we put in the other 3 to get LCM = 2*3*5*3 = 90 Ex. Find LCM for 6. 8.15 6 = 2*3 8=2*2*2 15=3*5 LCM start with 6 and get 2*3 Go to 8 and put in 2 3 s and get 2*3*2*2 Go to 15 and put in the 5 and get 2*3*2*2*5 LCM = 2*3*2*2*5 = 120 Greatest Common Factor The greatest common factor is the largest number that is a factor of each number in the set of numbers. It is used in the reduction of fractions. Finding the Greatest Common Factor Factor each number
Look at each factor in the first number and if it occurs in the other numbers (if it occurs in all numbers then include it in the GCF) Ex. Find the GCF of 8 and 12 8 = 2*2*2 12 = 2*2*3 GCF look at first 2 in 8 it is also in 12 so get 2 so far 8 = 2*2*2 12 = 2*2*3 now go to the next 2 of 8 it is also in 12 so we get 2*2 so far 8 = 2*2*2 12 = 2*2*3 now go to the next 2 of 8 it is not in 12 so we have 2*2 = 4 as the GCF Ex. Find the GCF of 60 and 200 60 = 2*2*3*5 200 = 2*2*2*5*5 look at things in common 60 = 2*2*3*5 200 = 2*2*2*5*5 so the GCF is 2*2*5 = 20
Reducing Fractions Whenever we are dealing with numbers in the terms of fractions, we like to have them reduced to lowest terms. The lowest terms of a fraction is the terms when the numerator and the denominator have no factors in common (relatively prime). Ex. 3 is in lowest terms since 3 and 4 are relatively prime 4 8 20 is not in lowest terms since they have 4 as a common factor To reduce fractions to lowest terms Factor numerator and denominator Cancel out factors in common Ex. Reduce 8 20 to lowest terms 2 2 2 2 becomes or 2 2 5 5 Ex. Reduce 44 100 to lowest terms 2 2 11 11 becomes or 2 2 5 5 25
Adding and Subtracting Fractions There will be occasions where will be necessary to add or subtract numbers that are fractions. (I know we don t like fractions, but they are necessary).to add fractions they must have the same denominator (bottom). If they do not have the same denominator then we must convert each of them to a fraction with a common denominator. Fractions with same denominators Ex. 1 2 1+ 2 3 + = =, just add the numerators (tops) 4 4 4 4 Ex. 4 2 4 2 2 = =, just subtract numerators 15 15 15 15 Fractions with different denominators. Ex. 1 1 +, different denominators. We must find a common multiple for the denominators to 2 3 use as a common denominator. The least common multiple of 2 and 3 is 6, so we use 6 as the common denominator. We convert each fraction to a new one with the denominator of 6. 1 1 3 3 3 = = multiply by 1 in the form of 2 2 3 6 3 1 1 2 2 2 = = multiply by in the form of 3 3 2 6 2 1 1 3 2 5 + = + = 2 3 6 6 6
Ex. 5 1 different denominators. The Least Common Multiple of 8 and 3 is 24 8 3 5 5 3 15 = = 8 8 3 24 1 1 8 8 = = 3 3 8 24 therefore 5 1 15 8 7 = = 8 3 24 24 24
Multiplying and Dividing Fractions Multiplication of fractions is very simple, just multiply numerators and denominators Ex. 1 3 1 3 3 = = 2 4 2 4 8 Ex. 2 3 6 1 = = 3 4 12 2 Division of fractions is not much harder but has one thing important to remember. You must invert the divisor and then multiply (Flip the last guy and multiply) Ex. 2 3 2 4 2 4 8 = = = 3 4 3 3 3 3 9 Ex. 15 5 15 4 60 3 = = = 8 4 8 5 40 2
Mixed Numbers and Improper Fractions There are two ways of expressing fractions representing numbers greater than one, mixed number and improper fractions. Mixed numbers are expressed as a whole number part and a fractional part in the form B 1 A like 3 C 2 Improper Fractions are fractions whose numerator is larger than the denominator A where A>B B Mixed Numbers to Improper Fractions To convert a mixed number to an improper fraction Multiply denominator by whole number part Add numerator Place over the denominator Ex. 1 Convert 3 to an improper fraction 2 Multiply denominator by whole number part 3*2 = 6 Add numerator 6 + 1 = 7 7 Place over the denominator 2 Ex. 3 Convert 5 to an improper fraction 4 5*4 = 20 20 + 3 = 23
23 so we get 4 Improper Fractions to Mixed Numbers To convert improper fractions to mixed numbers, we have to remember the long division that we learned in elementary school division with remainders. To convert from Improper Fractions to Mixed Numbers Ex. Ex. Perform implied division with remainder Write quotient as whole number part Place remainder over divisor as fractional part Write 19 as a mixed number 9 2 9 19R 1 so we get 1 2 9 Write 23 as a mixed number 4 5 3 4 23R3 so we get 5 4
Introduction to Decimals Numbers that cannot be represented as whole numbers are written as either fractions or in decimal notation. We are familiar with the concept of decimal notation from numerous examples in our lives, namely the use of money ($3.12 is decimal notation for 3 dollars and 12 cents) We can think of decimal notation as another way of writing certain special types of fractions (those with multiples of ten in the denominator) 3 Three tenths 0.3 Note: 1 zero in the denominator and 1 10 decimal place 3 Three hundredths 0.03 Note: 2 zeroes in the denominator and 2 100 decimal places 239 Two hundred thirty-nine 0.239 Note: 3 zeroes in the denominator and 3 1000 thousandths decimal places We should be able to note that there are exactly three parts to a decimal number. 3.12 whole number part decimal point decimal part Writing decimals numbers in words 0.03 is read as 3 hundredths since the 3 is in the second decimal place (1/100) 0.6481 is read as six thousand four hundred eighty-one ten-thousandths since the 1 is in the fourth decimal place (1/10000) Writing decimal numbers in standard form Five and thirty-eight hundredths hundredths implies a total of 2 decimal places to be filled by the 38 so we get 5.38 Nineteen and four thousandths thousandths implies a total of 3 decimal places to be filled by 4 so we add two leading zeroes to make 004 and get 19.004
Rounding decimals Sometimes we are called upon to limit the number of decimal places that can be used in a specific application (it makes no sense to take money out to 3 places). This process is known as Rounding. Rounding rules If the number to the right of the given place value is less than 5, drop that number and all numbers to the right of it. If the number to the right of the given place value is 5 or greater, increase the number in the given place value by one and drop all numbers to the right of it Round 26.3799 to the nearest hundredth Look at 26.3799, 7 is in the hundredths (second) place and 9>5 so increase 7 to 8 and drop the 99 and get 26.38 Round 42.0237412 to the nearest hundred thousandth Look at 42.0237412, 4 is in the hundred thousandth (fifth) place and 1 < 5 so drop the 12 and get 42.02374
Addition of decimals The addition of decimal numbers is almost identical to the addition of whole numbers. The only difference is that we need to remember to line up the vertical columns with respect to the decimal point. + 0.237 4.9 27.32 00.237 04.900 + 27.320 00.237 04.900 + 27.320 32.457 Add 0.237 + 4.9 + 27.32 Remember to line up on the decimal points Rewrite with zeroes added in appropriate place values to make things line up properly 7 + 0 + 0 = 7 3 + 0 + 2 = 5 2 + 9 + 3 = 14 carry 1 0 + 4 + 7 + (1) = 12 carry 1 0 + 0 + 2 + (1) = 3
Subtraction of decimals Subtraction of decimals is almost identical to subtraction of whole numbers. The only difference is to remember to line up the columns on the decimal point. (all rules of borrowing in subtraction apply) 6.93 3.7 6.93 3.70 6.93 3.70 3.23 Subtract 6.93 3.7 Add necessary zero to line up columns 3 0 = 3 9 7 = 2 6 3 = 3 Subtract 39.047 7.96 39.047 7.960 39.047 7.960 7 7 0 = 7 We need to borrow from the 9 (changed to 8) to change the 0 to 10 so that we can borrow from the 10(changed to 9) to change the 4 to 14 39 / 8.0/ 9 1 47 So the result is 31.087 7. 9 60 31. 0 87
Multiplication of Decimal Numbers Decimal numbers are multiplied in the same way as whole numbers, with special consideration given to the number of decimal places in each of the factors. (number of decimal places in first factor + number of decimal places in second factor = number of decimal places in product) Multiply 21.14 0.36 21.4 1 decimal place 0.36 2 decimal places 1284 642 1 + 2 = 3 7.704 3 decimal places Multiply 0.037 0.08 0.037 has 3 decimal places 0.08 has 2 decimal places 37 8 = 296 to make this number have 5 decimal places, we need to add 2 addition zeroes 0.00297 (the idea of keeping track of decimal places was necessary in the days when we used slide rules to perform our multiplications. The slide rule would do the whole number multiplication for us (3 8 = 297) but you had to put the decimal places in for yourself) Multiplication by multiples of ten To multiply by multiples of ten, move the decimal point to the left the same number of places as there are zeroes in the multiple of ten factor. 3.82 10 = 38.2 1 zero and 1 move to the left 3.82 100 = 382. 2 zeroes and 2 moves to the left 3.82 1000 = 3820. 3 zeroes and 3 moves to the left (needed to add a 0 at the right end to make the move)
Division of Decimal Numbers To divide decimal numbers, move the decimal point in the divisor to the right enough place to make it a whole number. Also move the decimal point in the dividend an equal number of places. (remember to keep the decimal point in the quotient directly above the decimal point in the dividend) Divide 3.25 15.275 We need to change 3.25 to 325 by moving the decimal point 2 places to the right Therefore, we need to change 15.275 to 1527.5 by moving the decimal point 2 places to the right. So the problem becomes 325 1527.5 4.7 325 1527.5 1300 2275 note: the decimal point in 4.7 is above the decimal point in1527.5 2275 0 Not all divisions of decimal numbers will come out even as in the above example. We generally round the quotient in a decimal division instead of carrying a remainder. Divide 0.3 0.56 and round to two decimal places First convert the problem to 3 5.6, to be able to round to two decimal places we must have 3 decimal places in the quotient so we will arrange to have 3 decimal places in the dividend 3 5.600 1.866 3 5.600 3 2 6 2 4 20 18 20 18 We will round 1.866 to 1.87
Dividing by multiples of ten To divide by multiples of ten, move the decimal point to the left the same number of places as there are zeroes in the multiple of ten divisor (placing leading zeroes as necessary) 34.65 10 = 3.465 1 zero in 10 so move 1 place left 34.65 100 = 0.3465 2 zeroes in 100 so move 2 places left 34.65 1000 = 0.03465 3 zeroes in 1000 so move 3 places left (needed to add a leading zero for the move)
Converting between Decimals and Fractions Fractions to Decimals To convert from a fraction to a decimal it is as simple as performing the implied division as we learned in elementary school. Terminating Decimals (division ends) Ex. 2.5 5 becomes 2.5 since 2 5.0 2.25 1 becomes.25 since 4 1.00 4 Non-terminating Decimals (repeating decimals) Ex..3333 1 becomes.3333 since 1 3.0000 repeating 3 s forever 3 Decimals to Fractions.1414 14 becomes.1414 since 99 14.00 repeating 14 s forever 99 Remember that decimals are actually fractions with the denominator as an appropriate power of ten (the number of zeroes after the one is equal to the number of places to the right of the decimal point) Ex. 5 1.5 has 1 place to right so it is reduced to 10 2 125 1.125 has 3 places right so it is reduced to 1000 8 34 2 3.4 has 1 place right so it becomes reduced to 3 10 5
Scientific Notation Occasionally, we need to deal with very large or very small numbers. It is convenient to use a system called scientific notation to represent these numbers. Scientific notation is based on powers of ten to represent these numbers. Large numbers 10 1 = 10 10 2 = 100 3 notice the exponent indicates the number of zeroes 10 = 1000 10 4 = 10,000 We can express large numbers as a number between 1 and 10 multiplied by the appropriate power of 10 289 = 2.89 10 2 notice the decimal point moved 2 units to the left (same as exponent) 36782 = 3.6782 10 4 4 place move and exponent of 4 Small Numbers 10-1 = 0.1 10-2 = 0.01 10-3 = 0.001 notice the exponent indicates the number of places right of decimal point 10-4 = 0.0001 We can express small numbers as a number between 1 and 10 multiplied by the appropriate power of 10. 0.234 = 2.34 10 1 we moved the decimal point 1 unit to the right 0.000123 = 1.23 10 4 we moved the decimal point 4 units right
Introduction to Ratios In real life, numbers are usually quantities of objects and have units associated with them. (6 balls, 12 feet and 9 cars) A ratio is a comparison between two numbers that have the same units. Ratios can be written in three different ways 3 feet 3 As a fraction =, ratios do not have units 4 feet 4 As two numbers separated by a colon 3 feet : 4 feet, 3 : 4 As two numbers separated by the word to 3 feet to 4 feet, 3 to 4 Compare two boards, one 6 feet long to another of 8 feet long. 6 feet 6 3 = =, this means that the shorter board is ¾ the length of the longer 8 feet 8 4 board
Introduction to Rates A rate is the comparison of two quantities that have different units. Rates are written as fractions A cyclist rode 144 miles in 10 hours. We can write a distance to time ratio for the trip 144 miles 72 miles =, always simplify the fraction, whenever possible 10 hours 5hours Unit rates Whenever a rate has 1 as the denominator, it is referred to as a unit rate. $6.25 $3.25 2 pounds of sirloin sell for $6.25, =, or $3.25 per pound 2lbs 1lb 360 miles 60 miles A car travels 360 miles in 6 hours, =, 60 miles per hour 6hours 1hour
Introduction to Proportions A proportion is an expression of the equality of two ratios or rates. 50 miles 25 miles = is a proportion 4gals 2gals A proportion is true if the two fractions are equal (when written in lowest terms). The best way to check if a proportion is true is to check the equality of the cross products a c.i.e. given = is true if ad = bc b d 2 8 Is = a true proportion? 3 12 2*12 = 24 3*8 = 24 24 = 24 so it is a true proportion A proportion is false if the cross products are not equal. 4 8 Is = a true proportion? 5 9 4*9 = 36 5*8 = 40 therefore a false proportion Solving proportion problems Sometimes, we do not know one of the values of a proportion. We use the above mentioned property of cross products to solve for the missing value 9 3 Solve = 6 n By cross multiplying we get 9n = 6(3) 9n = 18 9n 9 = 18 9 n = 2 9 3 thus = is a true proportion 6 2
Introduction to Percents Percents, like fractions and decimals, are ways of writing numbers that are not necessarily whole numbers. Percent means parts of 100, so they can always be written as fraction with 100 in the denominator. Writing percents as a fraction 1 To write percents as a fraction, multiply the percent by 100 1 13 1 13 13% = 13 = = 100 1 100 100 1 120 1 120 6 1 120% = 120 = = = = 1 100 1 100 100 5 5 2 2 1 50 1 50 1 16 % = 16 = = = 3 3 100 3 100 300 6 Sometimes it helps to change mixed numbers to improper fractions for the calculation Writing percents as decimals To write a percent as a decimal, multiply the percent by 0.01 (move decimal point two places to the right) 5% = 5 * 0.001 = 0.05 215% = 215 * 0.001 = 2.15 Writing fractions or decimals as percents 100% To write fractions or decimals as percents, multiply the fraction or percent by Change3/8 to a percent 3 3 100 300 100% = % = % = 37.5% 8 8 1 8 Change 0.48 as a percent 0.48 * 100% = 48% Table of Common Fractions and Their Percentage Equivalents 1 / 2 = 50%
1 / 3 = 33 1 / 3 % 2 / 3 = 66 2 / 3 % 1 / 4 = 25% 3 / 4 = 75% 1 / 5 = 20% 2 / 5 = 40% 3 / 5 = 60% 4 / 5 = 80% 1 / 6 = 16 2 / 3 % 5 / 6 = 83 1 / 3 % 1 / 7 = 14 2 / 7 % 2 / 7 = 28 4 / 7 % 3 / 7 = 42 6 / 7 % 4 / 7 = 57 1 / 7 % 5 / 7 = 71 3 / 7 % 6 / 7 = 85 5 / 7 % 1 / 8 = 12 1 / 2 % 3 / 8 = 37 1 / 2 % 5 / 8 = 62 1 / 2 % 7 / 8 = 87 1 / 2 % 1 / 9 = 11 1 / 9 % 2 / 9 = 22 2 / 9 % 4 / 9 = 44 4 / 9 % 5 / 9 = 55 5 / 9 % 7 / 9 = 77 7 / 9 % 8 / 9 = 88 8 / 9 % 1 / 10 = 10% 3 / 10 = 30% 7 / 10 = 70% 9 / 10 = 90% 1 / 12 = 8 1 / 3 % Table of Common Fractions and Their Decimal Equivalents or Approximations 1 / 2 = 0.5 1 / 3 = 0.3333... 2 / 3 = 0.6666... 1 / 4 = 0.25 3 / 4 = 0.75 1 / 5 = 0.2 2 / 5 = 0.4 3 / 5 = 0.6 4 / 5 = 0.8 1 / 6 = 0.1666... 5 / 6 = 0.8333... 1 / 7 = 0.142857142857... 3 / 7 = 0.428571428571... 5 / 7 = 0.714285714285... 2 / 7 = 0.285714285714... 4 / 7 = 0.571428571428... 6 / 7 = 0.8571428571428... 1 / 8 = 0.125 3 / 8 = 0.375 5 / 8 = 0.625 7 / 8 = 0.875 1 / 9 = 0.111... 2 / 9 = 0.222... 4 / 9 = 0.444... 5 / 9 = 0.555... 7 / 9 = 0.777... 8 / 9 = 0.888... 1 / 10 = 0.1 3 / 10 = 0.3 7 / 10 = 0.7 9 / 10 = 0.9 1 / 12 = 0.08333...
Percent Equations There are many ways to solve problems involving percents. All of them require that you identify three parts of the problem (percent, base and amount), but most of require that you also learn rules for when you multiply and when you divide. This is fairly confusing. The best way to solve the basic percent problem is to use the proportion, percent amount = and the cross multiplication, percent * base= 100 * amount. 100 base 4% of 85,000 is what number? First, we must determine where the above numbers go in the proportion. As a amount general rule, you can look at the ratio by looking at the wording of the base problem. The number associated with the word of generally as the base and the amount is number associated with the word is generally is the amount. So, =. base of 4 = percent is = what number (the variable (n)) of = 85,000 4 n =, by cross multiplying we get 100 85,000 4(85,000) = 100n 340000 = 100n 3400 = n What percent of 40 is 30? Is = 30 Of = 40 Percent = what (n) n 30 = 100 40 40n = 3000 n = 75% 18% of what is 900? Percent = 18 Is = 900 Of = what (n) 18 900 = 100 n 18n = 90000 n = 5000
Applications of Percents A certain charitable organization spent $2940 for administrative costs. This is 12% of the total amount of the monies they collected. How much did they collect in total? We are really asking $2940 is 12% of what number? 12 2940 = 100 n 12n = 294000 n =24,500 so, they collected a total of $24,500 The fire department reports that 24 false alarms were sent in out of a total of 200 alarms. What percent of all the alarms are false? We are really asking 24 is what percent f 200? n 24 = 100 200 200n = 2400 n = 12% so, 12% were false alarms An antiques dealer claims that 86% of her sales are for items that cost less than $1000. If she sells 250 items in a given month, how many will sell for less than $1000? We are really asking what number is 86% of 250? (notice $1000 is not used anywhere) 86 n = 100 250 100n = 21500 n = 215 so, 215 sold for less than $1000
Negative Numbers There are special numbers associated with the positive numbers that are called the additive inverses. These are numbers that when added to a positive number will give 0 as the result. Ex. 4 + something = 0 something = -4, therefore 4 + -4 = 0 6.6 + -6.6 = 0 1024 + -1024 = 0 These additive inverses of positive numbers are called negative numbers. Many of us are familiar with negative numbers when we look at our checking accounts (the US Government).
Adding and Subtracting Signed Numbers There are basic rules used in the addition and subtraction of signed (+ or -) Addition of signed numbers There are two situations to consider in the addition of two signed numbers (or the signs the same or different) Signs the same If the signs of the two numbers are the same, just add the numbers and apply the common sign Ex. 4 + 5 = 9 Both are positive so 9 is positive (-4) + (-5) = (-9) Both are negative so -9 is negative Signs different If the signs of the two numbers are different then subtract the two numbers and use the sign of the number with the larger absolute value Ex. 4 + (-5) 5 4 = 1 and since 5 > 4 we use the from -5 to get -1-41 + 51 51 41 = 10 and 51 > 41 so we get +10 (10) Subtraction of Signed Numbers The best way to subtract signed numbers is to not do it. You can always convert a subtraction into an addition and work from there. Ex. Ex. 4 5 convert to 4 + (-5), the conversion to to addition requires a sign change in the number after the subtraction sign 4 + (-5) = -1 by addition rules above -4 5 convert to -4 + (-5) = -9
rules Ex. 4 (-5) convert to -4 + 5 (note -5 changed to 5) so -4 + 5 = 1 by addition When it comes to larger, more complicated problems, remember you can only add or subtract two numbers at a time, so only work with two at a time Ex. 4 + (-5) 6 + (-7) 9 pretty long, but take two at a time 4 + (-5) = -1 so we get -1 6 + (-7) 9-1 6 = -1 + (-6) = -7 so we get -7 + (-7) 9-7 + (-7) = -14 so we now get -14 9 = -14 + (-9) = -23
Multiplying and Dividing Signed Numbers There are exactly two rules that must be followed when multiplying or dividing two signed numbers If the signs agree, the answer is positive If the signs disagree, the answer is negative Remember, the operations of multiplication and division can only be done with 2 numbers at a time complicated problems must be taken two numbers at a time. Ex. 4*(-5) = -20 opposite signs (-4)*(-5) = 20 same signs 20 = 5 4 20 = 4 5 opposite signs same signs Ex. (-4)(-5)(2)(-6) (-4)(-5) = 20 20(2)(-6) (20)(2) = 40 40(-6) -240
Variables Throughout our study of algebra, we hear about these things called variables. What is a variable? Variables are just symbols to represent values that are currently unknown. They stand in the place of numbers. Ex. In the expression 3x, x represents a number and is called a variable In the expression 2x -4y, x and y both represent different numbers and are called variables Throughout algebra, and actually real life, we encounter unknown values that can be expressed as variables. If you go to the grocery store and find that 3 cans of cream corn cost $1.29, the price of one can is unknown and can be represented by a variable (maybe p for price)
Evaluating Algebraic Expressions Algebraic expressions will have a value dependent upon the number that is substituted into the variable. Example: Evaluate 4x + 3 if x = 1 4(1) + 3 = 7 + 3 = 10 Evaluate 4x + 3 if x = -2 4(-2) + 3 = -8 + 3 = -5
Combining Like Terms in Algebraic Expressions Two or more terms in an algebraic expression that have the same variable part are called like (similar) terms. (Think of apples and oranges. If an expression is made up of 3 apples and 2 oranges, they cannot be combined since apples oranges) If an algebraic expression contains 2 or more like terms, they can be combines to form a single term. Examples: 4x + 7x = (4 + 7) x = 11x 13u 7u + 2u = (13 7 + 2) u = 8u 6x + 3x + 7y = (6 + 3) x + 7y = 9x + 7y Not like terms
Properties of Equations The major thing to remember when working with equations, is when you make a change to one side of the equation you must make an identical change to the other side. Rule: You can add or subtract a value to both sides of an equation without changing the solution to the equation. Examples; X + 3 = 7, if I add 3 to the left, I must add 3 to the right X + 3 + 3 = 7 + 3 X + 6 = 10 Y + 6 = 10, if I subtract 6 from the left, I must subtract 6 from the right Y + 6 6 = 10 6 Y = 4 Rule: You can multiply both sides of the equation by a constant or divide a constant into both sides of the equation without changing the solution to the equation. Examples: -4c = 20 I can divide both sides by 4-4c -4 = 20-4 c = 5 2 5 x = 3 9 I can multiply both sides by 3/2 3 2 3 5 x = 2 3 2 9 2/3 * 3/2 =1 15 5 x = = 18 6 You can also combine both of the above rules in the same problem to simplify. 3x + 2 = 8 Subtract 2 from both sides 3x + 2 2 = 8 2 3x = 6 Divide both sides by 3 3x 3 = 6 2 x = 2
1 x 2 = 6 2 1 x 2 + 2 = 6 + 2 2 1 x = 8 2 1 x 2 = 8 2 2 x = 16 Add 2 to both sides Multiply both sides by 2
Algebraic Equations When we connect two algebraic expressions using =, we call the combination an equation. There are three parts to an equation X + 3 = 7 Left side equal sign right side An equation can be true or false, dependent upon the number substituted into the variable. Examples X + 3 = 7 is true if X = 4 X + 3 = 7 is false if X = 1 Some equations are true for any value of the variable. These types of equations are called identities. Example: 2(3x + 7) = 6x + 14 is an identity x = 5 2(3(5) + 7) = 6(5) + 14 44 = 44 x = 1 2(3(1) + 7) = 6(1) + 14 20 = 20 Most equations are not identities. They are true for only certain values of the variable. These are called conditional equations. Example: 5x 2 = 3 is true if x = 1 5x 2 = 3 is false if x = any other number 5x 2 = 3 is a conditional equation The process of determining the values that make a conditional equation true is called solving the equation.
Solving Linear Equations Linear Equation An equation where the only exponent involved with the variable is 1 is called a linear equation. Examples: 3x + 2 = 8 2(x + 4) x + 7 = 6 are both linear equations The best way to look at the procedures for solving a linear equation is to undo the operations that have been done to the variable. Example: Equation Done to x How undone 3x + 2 = 8 2 added to 3x Subtract 2 3x + 2 2 = 8 2 3x = 6 Multiplied by 3 Divide by 3 3x 3 = 6 2 x =2 We also need to be able to get the variables on one side and the constants on the other sides. Example: 3x + 2 = x 6 Move x to left by subtracting x from both sides 3x + 2 x = x 6 + x 2x + 2 = 6 Subtract 2 from both sides 2x + 2 2 = 6 2 2x = 4 Divide both sides by 2 2x 2 = 4 2 x = 2 Procedures for Solving Linear Equations Done 1. Perform any implied operations (multiply through parenthesis, etc.) 2. Combine like terms on each side 3. Perform necessary operations to separate constants and variables 4. Combine like terms as necessary 5. Multiply or divide to make the numerical coefficient 1
Literal Equations Literal equations are also called formulas. They form a relationship between two or more variable, such as the formula to find the area of a circle from the radius A = π r 2. Occasionally, we will need to solve for the variable that is not explicitly solve for in the equation, like trying to find r from the formula above. To solve for a variable in a formula, treat the other variables as constants and solve as you normally would any other equation. Ex. Solve A = 2L + 2W (area of a rectangle) for L A = 2L + 2W subtract 2W from each side A 2W = 2L divide both sides by 2 (A-2W)/2 = L given A and W can find L Solve C = 2 r for r (circumference of a circle) C = 2 r divide both sides by 2 (remember is just a number) C/2 = r can find r given C
Linear Inequalities The rules for solving linear inequalities are the same as those for linear equalities except for one thing, if you multiply or divide the inequality by a negative number the inequality sign switches. The solutions to linear inequalities are intervals of numbers, not individual numbers like in equalities. Interval Notation There are 2 symbols used to show the endpoints of intervals Ex. ( - endpoint of the interval not included [ - endpoint of interval included (0,2) all numbers between 0 and 2 but not including 0 or 2 [0,2) all numbers between 0 and 2 including 0 but not 2 [0,2] all numbers between 0 and 2 including 0 or 2 (-, 2] all numbers less than or equal to 2 (2, ) all numbers greater than 2 but not 2 Solving Inequalities Ex. Solve 3x + 2 > 0 subtract 2 from each side 3x > -2 divide by 3 x > -2/3 (-2/3, ) Solve -4x 8 < 12 add 8 to each side -4x < 20 divide by -4(remember to switch sign) x > -5 (-5, )
Solve 3x + 4 < 2x 1 put x s on one side and constants on the other 3x + 4 < 2x 1 subtract 4 from each side 3x < 2x 5 subtract 2x from each side x < -5 (-, -5)
Multiplication using Exponents Exponents are a shortcut form for repeated factors in a multiplication. Examples; 3 4 = 3*3*3*3 = 81 (-2)5 = (-2)*(-2)*(-2)*(-2)*(-2) = -32 2 2 2 2 8 ( ) 3 = = 3 3 3 3 27 x n = x x x x n times Parts of the exponential number x n Base Exponent There are a few rules governing the use of exponential numbers Rule 1 Rule 2 Rule 3 Multiplication Property of like terms a m * a n = a m+n 3 4 * 3 7 = 3 4+7 = 3 11 Power to a Power Property (a m ) n = a mn (4 2 ) 6 = 4 2*6 = 4 12 Power of a Product Property n b n (ab) n = a (5x) 2 = 5 2 x 2 = 25x 2 (-x) 5 = (-1) 5 x 5 = -x 5 (-x) 6 = (-1) 6 x 6 = x 6
Further Examples: (2x 3 ) 4 = 2 4 x 3*4 = 24x 12 = 16x 12 3 ) 3 (-3xy 2 ) 2 (2x 3 y 2 2*2 2 3 3 * 3 3*3 (-3) 2 x y x y 9x 2 y 4 8x 9 y 9 72 x 11 y 13
Division with Exponents n a m In order to evaluate we need to develop a few rules that will lead us to a some a interesting properties of exponents. Look at note:5 3 = 2 Therefore we have a rule n a m a n m = a Examples: 6 5 6 4 2 = 5 = 5 4 5 6 4 0 = 6 6 4 6 4 also = 1 therefore 6 0 = 1 4 6 0 From here, we get a = 1 Zero Power Rule 5 4 4 6 2 = 5 = 5 5 6 5 4 5*5*5*5 1 1 1 also = = = therefore 5 2 = 6 2 2 5 5*5*5*5*5*5 5*5 5 5 leading us to the a n = 1 a n Negative Power Rule : giving us 3 2 3 3 3 ( ) = ( )( ) = 4 4 4 4 a ) b a b n ( n = n 2 2 = 16 9 Power of a Quotient Rule
Further Examples: 4 4*5 20 x x x ( ) 5 = ( ) = 3 3*5 15 y y y 2 4 3 2 12 2 12 6 9x ( y ) 9x y 9 x y 1 6 y = = = 1 y = 2 3 2 4 6 4 6 2 2 (3x y ) 9x y 9 x y x x note: the variable ends up on the side of the fraction line as the largest power 6 3 2 2 3* 2 2* 2 6 4 x (x y ) = x y = x y = 4 y note: the negative power causes the variable to change sides of the fraction line
Polynomials Polynomials are a special type of algebraic expression where the exponents are positive integers. Ex. x 2 + 2x -1 is a polynomial since exponents are 2 and 1 x 3/2 is not a polynomial since exponent is 3/2 x is not a polynomial (radical variables have fractional exponents) 3x is a polynomial since exponent is 1
Evaluating Polynomials Polynomials will have a value dependent upon the number that is substituted into the variable. Example: Evaluate 4x + 3 if x = 1 4(1) + 3 7 + 3 10 Evaluate 4x + 3 if x = -2 4(-2) + 3-8 + 3 = -5 Evaluate y 3 + 3y 5 if y = -3 (-3) 3 + 3(-3) 5-27 - 9 5-40 Evaluate y 3 + 3y 5 if y = 0 (0) 3 + 3(0) 5 0 + 0 5-5 Evaluate 3u 2 + 2uv 12 if u = -1 and v = 5 3(-1) 2 + 2(-1)(5) 12 3(1) 10 12-19
Adding Polynomials Adding polynomials is basically the same as the addition of numbers with units. You can only add terms that are similar. Ex. 2 apples + 3 oranges + 5 apples = (2 + 5) apples + 3 oranges = 7 apples + 3 oranges (you can t add apples and oranges) Ex. 2x + 3x = (2 + 3)x = 5x Ex. 5y + 3x + 8y + 4x = (5 + 8)y + (3 + 4)x = 13y + 7x Ex. 2x 2 +3x 2 + 4x 3 = 5x 2 + 4x 3 (note:x 2 and x 3 are not similar terms)
Subtracting Polynomials The subtraction of polynomials is basically the same as the addition of polynomials, with one exception (you have to deal with the subtraction sign). We accomplish this by remembering how to subtract signed numbers change the subtraction to addition and change the sign of everything after the minus sign. Ex. (3x 2 + 2x + 4) (-2x 2 + 3x + 1) change to (3x 2 + 2x + 4) + (2x 2 3x 1) combine like terms (3x 2 +2x2) + (2x 3x) + (4 1) perform operations 5x 2-1x + 3 Ex. (3xy 4y + 3xz) (3y -3xy +xz) (3xy - 4y + 3xz) + (-3y + 3xy xz) change to combine like terms (3xy +3xy) + (-4y +3y) + (3xz xz) perform operations 6xy y + 2xz
Multiplying Polynomials Multiplying polynomials is mostly a trial in keeping things lined up and not missing any parts of the problem. Technique to multiply polynomials Break into several simple problems using distribution Multiply each part Combine like terms Ex. (3x 2 + 2)(3x 1) use distribution of first term 3x 2 (3x -1) + 2(3x 1) distribute again 3x 2 (3x) -3x2(1) + 2(3x) 2(1) Multiply terms 9x 3 3x 2 + 6x -2 Ex. (6x +2y + 1)(2x -3y 2) 6x(2x -3y 2) + 2y(2x 3y 2) + 1(2x -3y 2) 6x(2x) 6x(3y) -6x(2) +2y(2x) 2y(3y) -2y(2) +1(2x)-1(3y)-1(2) 12x 2-18xy-12x + 4xy 6y 2-4y + 2x 3y -2 12x 2-18xy +4xy -12x +2x -4y -3y -6y 2-2 12x2-14xy -10xy -7y - 6y 2 2 FOIL Method If you are multiplying a binomial by a binomial you can use a variation of the above technique call First Outside Inside Last (FOIL) (ax + b)(cx + d) Ex. First Outside Inside Last (x + 2)(x + 3) First Outside Inside Last (ax)(cx) = acx 2 ax(d) b(cx) bd acx 2 +(ad + bc)x + bd x(x) x(3) 2(x) 2(3) x 2 +3x +2x + 6 x 2 + 5x + 6
Special Products There are a few special types of products that are handy to keep in your back pocket for use in a hurry. These are not necessary to memorize, but they can help you perform multiplication of polynomials rapidly, in the cases where they apply. Special Product #1 Special Product #2 Special Product #3 2 2 2 (a + b) = a + 2ab + b Example 2 2 2 2 (6 + y) = 6 + 2(6)( y) + y = 36 +12 y + y 2 2 2 (a b) = a 2ab + b Example: 2 2 2 2 (6 y) = 6 2(6)( y) + y = 36 12 y + y (a + b)( a b) = a 2 b 2 Example: (3 + c)(3 c) = 3 2 c 2 Further Examples using the Special Product Rules 2 2 2 2 (8 + 2y) = 8 + 2(8)(2y) + (2y) = 64 + 32 y + 4y 3 2 1 2 3 2 2 3 2 1 1 2 6 4 3 2 1 (4m n ) = (4m n ) 2(4m n )( ) + ( ) = 16 m n 4m n + 2 2 2 4 2 2 2 3 2 3 2 3 4 6 (3x 2 y )(3x + 2y ) = (3x ) (2 y ) = 9x 4 y
Dividing Polynomials There are two types of division problems for polynomials, those that can be done as fractions and those that require long division. If the divisor is a single value then the division can be broken into a group of simple fractions, otherwise you must use long division. Division using fractions Ex. 18 x 2 4x 18 x 2 = 4x = 9x 2 2x 2x 2x Long Division Ex. 2 x + 1 x + 2x + 1 x 2 x + 1 x + 2x + 1 (x 2 + x) x +1 x 2 divided by x gives us x multiply x(x+1) and subtract x divided by x gives 1 x + 1 2 x + 1 x + 2x + 1 (x 2 + x) x +1 (x +1) 0 Multiply 1(x + 1) and subtract Therefore, we get (x 2 + 2x + 1) (x + 1) = x + 1
Common Factors We can expand the idea of the Greatest Common Factor, that is studied in earlier classes into the world of algebra. We will be looking for the common factors in algebraic expressions. Look at 30x 2 and 42x 3. We can factor them individually. 30x 2 = 2 * 3 * 5 * x * x 42x 3 = 2 * 3 * 7 * x * x * x Each term has a 2, a 3, and 2-x s so they have 2 * 3 * x * x = 6x 2 in common 30x 2 = 6x 2 (5) 42x 3 = 6x 2 (7x) Example: Find the common factors for 15x 3 y 3, 6x 2 y 3 and 9xy 4 First look at the coefficients 15, 6, and 9 The common factor of 15, 6, and 9 is 3 Look at the x variables The exponents on x are 3, 2, and 1 choose the lowest number, x 1 = x is the common x term Look at the y variable The exponents of y are3, 3, and 4 choose the lowest number, x 3 is the common term. Therefore the common term is 3xy 3 15x 3 y 3 = 3xy 3 (5x 2 ) 6x 2 y 3 = 3xy 3 (2x) 9xy 4 = 3xy 3 (3y 3 )
Factoring polynomials by common factors A polynomial is factored when it is expressed as a product of 2 or more polynomials. The distributive property of integers and algebra, P(Q + R) = PQ + PR is the greatest aide in factoring. For the purposes of factoring we generally write the property in reverse order PQ + PR = P(Q + R). In this way we can see that P and (Q +R) are the factors of PQ + PR. Example: Factor 5x + 30 by common factors Since 5x = 5 * x and 30 = 5 * 6, 5 is the common factor 5x + 30 = (5 * x) + (5 * 30) 5x + 30 = 5(x + 30) by reverse distributive law. Procedure to factor by common factors. 1. Find the common factor in all the terms 2. Write each term as a product using the common factor found in step 1 3. Use reverse distributive law to factor out the common factor Example: Factor 5y 3 + 25y The common factor in the terms is 5y 5y 3 + 25y = 5y(y 2 ) + 5y(5) so 5y 3 + 25y = 5y(y 2 + 5) Example: Factor 18h 5 + 12h 4 21h 3 The common factor is 3h 3 18h 5 = 3h 3 (6h 2 ) 12h 4 = 3h 3 (4h) 21h 3 = 3h 3 (7) so 18h 5 + 12h 4 21h 3 = 3h 3 (6h 2 ) + 3h 3 (4h) - 21h 3-3h 3 (7) 18h 5 + 12h 4 21h 3 = 3h 3 (6h 2 + 4h 7)
Example: Factor 4a 7 b 5 c + 10a 3 b 8 16a 4 b 6 + 18a 5 b 7 The common factor is 2a 3 b 5, note c is not in all terms 4a 7 b 5 c = 2a 3 b 5 (2a 4 c) 10a 3 b 8 = 2a 3 b 5 (5b 3 ) 16a 4 b 6 = 2a 3 b 5 (8ab) 18a 5 b 7 = 2a 3 b 5 (9a 2 b 2 ) so 4a 7 b 5 c + 10a 3 b 8 16a 4 b 6 + 18a 5 b 7 = 2a 3 b 5 (2a 4 c) + 2a 3 b 5 (5b 3 ) - 2a 3 b 5 (8ab) + 2a 3 b 5 (9a 2 b 2 ) 4a 7 b 5 c + 10a 3 b 8 16a 4 b 6 + 18a 5 b 7 = 2a 3 b 5 (2a 4 c + 5b 3-8ab + 9a 2 b 2 )
Factoring by Grouping Certain polynomials, though they have no common factors, can be still be factored by using common factors. We have to be able to re-group the terms so that they are in groups that have common terms. Look at 3xm + 3ym 2x 2y 2 terms have x and 2 terms have y (3xm 2x) + (3ym 2y) Factor x out of the first part and y out of the last part x(3m 2) + y(3m 2) note: 3m 2 is now in common (x + y)(3m 2) Example: Factor 3a + 3b ma mb (3a + 3b) + (-ma mb) Factor 3 out of first part and m out of second part 3(a + b) m(a + b) note: factoring m out of mb leaves +b (3 m)(a + b)
Solving Equations by Factoring We can use factoring to solve algebraic equations. We will use the property of multiplication of two numbers equaling zero, (if a*b = 0 then either a = 0 or b = 0) Solve x 2 + 5x = 0 By factoring we get x(x + 5) = 0 therefore x = 0 or x + 5 = 0 which implies x = 0 or x = -5 Solve 2x 2 6x = 0 By factoring we get 2x(x 3) = 0 therefore 2x = 0 or x 3 = 0 which gives us x = 0 or x = 3 Solve x2 +5x + 6 = 0 By factoring we get (x + 3)(x + 2) = 0 therefore x + 3 = 0 or x + 2 = 0 which leads to x = -3 or x = -2
Factor the difference of two squares We know from our multiplication rules that (a + b)(a b) = a 2 b 2. Applying this rule backwards we get a factoring rule for the difference of two squares, a 2 b 2 = (a + b)(a b) If we can identify the problem as being the difference of two squares, we can apply above rule. Example: Factor x 2 4 x 2 2 2 (x + 2)(x 2) a = x and b = 2 Example: Factor 4t 2 9s 2 (2t) 2 (3s) 2 a = 2t and b = 3s (2t + 3s)(2t 3s) Example: Factor u 4 16 (u 2 ) 2-4 2 a = u 2 and b = 4 (u 2 + 4)(u 2 4) (u 2 + 4)(u 2 2 2 ) a = u and b = 2 (u 2 + 4)(u + 2)(u 2)
Factoring the sum or difference of cubes If a polynomial is made up of the sum of difference of two perfect cubes, we have special rules to handle the factoring (these are not easy to memorize, I can never remember them without looking). a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Example: Factor x 3 + 8 note: 8 = 2 3 Therefore, we get x 3 + 2 3 a = x and b = 2 (x + 2)(x 2 2x +2 2 ) (x + 2)(x 2 2x +4) Example: Factor 64u 3 27v 6 note: 64u 3 = (4u) 3 and 27v 6 = (3v 2 ) 3 (4u) 3 - (3v 2 ) 3 a = 4u and b = 3v 2 (4u 3v 2 )((4u) 2 + (4u)(3v 2 ) + (3v 2 ) 2 ) (4u 3v 2 )(16u 2 + 12uv 2 +3v 4 )
Introduction to Rational Expressions A rational expression is a fraction (p/q) where p and q are polynomials. 3 2y 7c +1 Ex.,, are all rational expressions x y +1 c 2 1 Evaluation of Rational Expressions As with polynomials the value of a rational expression is dependent upon the value chosen for the variable. x + 2 Ex. Evaluate for x = -1, x = 2, and x = 1 x 1 1+ 2 1 1 x = -1 gives us = = 1 1 2 2 2 + 2 4 x = 2 gives us = = 4 2 1 1 1+ 2 3 x = 1 gives us = which implies there is no solution 1 1 0 It is important to realize the denominator (bottom) of a fraction can not equal to zero. Zero in the denominator of a fraction causes the fraction to be undefined. x 5 Ex. For what value of x is undefined? 3x +1 For a rational expression to be undefined, the denominator must be equal 1 to zero therefore we set 3x + 1 = 0 and get x = -. Thus we cannot 3 1 x 5 substitute - into the fraction 3 3x +1
Equivalent Rational Expressions If two rational expressions can be reduced to the same rational expression, then the two rational expressions are called equivalent rational expressions 6x 2 Examples: and are equivalent fractions. 9x 3 6 y 2 y and are not equivalent 2y + 6 y + 3 2y is in reduced form y + 3 3y 2y y + 3 y + 3 Equivalent fractions in signed number systems p p p = = q q q p p p p = = = q q q q Examples: 1 1 1 = = y x ( y x) x y a b (b a) b a = = x y ( y x) y x
Simplifying Rational Expressions As with the more familiar numerical fractions we can reduce rational expressions by canceling common factors in the numerator and the denominator. pk = p Examples; Reduce 8 x 12 x qk q 12x 2 y Reduce 9xy 2 4x 2 1 Reduce 2x 2 + x Warning: it is common to try factoring improperly. Remember, you can only factor out factors that in a product, not terms in and addition or subtraction Proper reducing Improper reducing
Addition and Subtraction of Rational Expressions As in numerical fractions, we can only add and subtract rational expressions (algebraic fractions) if they have like (common) denominators. a c a + c + = b b b a c a c = b b b Examples: 2x 3x 2x + 3x 5x + = = 7 7 7 7 If the two fractions do not have a common denominator, we must convert the individual fractions, separately, to fractions with the same denominator(lcd). Example: 2x 4x Add +, the denominators are 3 and 5, so the LCD is 15 3 5 2x 2x 5 10x 4x 4x 3 12x = = and = = therefore we get 3 3 5 15 5 5 3 15 10x 12x 22x + = 15 15 15 Finding the Least Common Denominator We create a LCD in algebraic fractions just like we did in arithmetic fractions 1. Factor each denominator completely 2. List each prime factor the greatest number of times that it appears in either of the factored forms of the denominator. 3. The product of these factors is the LCD.
Examples: therefore 7 8 10x = 2*5* x + 15x = 3*5* x 10x 15x LCD = 2*3*5* x = 30x 7 7 3 21 = = 10x 10x 3 30x 8 8 2 16 = = 15x 15x 2 30x 21 16 37 + = 30x 30x 30x 5 7 18x = 2 3 3 x + 12xy = 2 2 3 x y 18x 12xy LCD = 2 2 3 3 x y = 36xy 5 5 2y 10y = = 18x 18x 2y 36xy therefore 7 7 3 21 = = 12xy 12xy 3 36xy 10y 21 10y + 21 + = 36xy 36xy 36xy therefore 3 4 a 2 b b 3 c 3 3 b 2 c 3b 2 c = = 2 2 2 2 3 a b a b b c a b c 4 4 a 2 4a 2 = = 3 3 2 2 3 b c b c a a b c 3b 2 c 4a 2 a 2 b 3 c LCD = a 2 b 3 c do not simplify at this point
1 4 + LCD = ( y + 3)(y + 4) y + 3 y + 4 1 1 y + 4 y + 4 = = y + 3 y + 3 y + 4 (y + 3)(y + 4) therefore 4 4 y + 3 4(y + 3) 4y +12 = = = y + 4 y + 4 y + 3 (y + 3)(y + 4) ( y + 3)(y + 4) y + 4 4y +12 5y +16 + = ( y + 3)(y + 4) ( y + 3)( y + 4) (y + 3)(y + 4)
Multiplication and Division of Rational Expressions Multiplication of Rational Expressions Multiplication of Rational Expressions is primarily the same as multiplication with numerical fractions. Examples: Division of Rational Expressions Division of Rational Expressions involves using the old adage flip the last guy and multiply Invert and Multiply Examples: 4 3t = 4 7t = 28t = 28 25 7t 25 3t 75t 75 Note the inverting 3 3 2 3 2 2 16a 8a 16a 25b 400a b 10a = = = 3 2 3 3 5b 25b 5b 8a 40ab b
Complex Fractions A complex fraction is one in which the numerator or denominator or both contain fractions. Examples: 5 16 7, x 9, 3 11 x x x 3 are all complex fractions To simplify a complex fraction, we express it as a simple fraction of lowest terms. The best way to handle a complex fraction is to realize that fractions are really quotients. Then you can use the old adage Flip the last guy and multiply (Invert and multiply) Examples: 3 4 3 3 3 5 15 5 = = = = 3 4 5 4 3 12 4 5 3 1 + 4 3 1 1 + 3 6 Hint: confront the top and bottom separately 3 1 9 4 13 + = + = 4 3 12 12 12 13 12 13 2 13 = = 1 12 1 6 2 1 1 2 1 3 1 + = + = = 3 6 6 6 6 2
1 x x remember to handle top and bottom separately 1 1+ x 1 x 1 x 2 1 x 2 1 x = = = x 1 x x x x 1 1 1 x 1 x +1 1+ = + = + = x 1 x x x x x 2 1 x x 2 1 x (x +1)(x 1)x = = = x 1 x +1 x x +1 x(x +1) x
Equations involving Algebraic Fractions We have to occasionally solve equations involving fractions. We have a procedure to follow to help us in this quest. Solve equations involving fractions 1. Determine the LCD 2. Multiply both sides of the equation by the LCD (this step eliminates the fractions) 3. Solve the resulting equation 4. Check you solution. Sometimes the solution to the resulting equation will not be a solution to the original algebraic problem (it will cause the denominator of the fraction to be zero) Examples: t 2t 55 + = LCD = 12 4 3 12 t 2t 55 12( + = ) 4 3 12 t 2t 55 12( ) +12( ) = 12( ) 4 3 12 3t + 8t = 55 11t = 55 t = 5 1 1 5 1 + = +, LCD = 6y y 2 6 y 3 1 1 5 1 6y( + = + ) y 2 6y 3 1 1 5 1 6y( ) + 6y( ) = 6y( ) + 6y( ) y 2 6y 3 6 + 3y = 5 + 2y 3y 2y = 5 6 y = 1 since y = -1 will not cause any of the denominators to become zero, it is the solution
w 2 + 2 =, LCD is w-2 w 2 w 2 w 2 (w 2)( + 2 = ) w 2 w 2 w 2 (w 2)( ) + (w 2)2 = (w 2)( ) w 2 w 2 w + 2w 4 = 2 3w = 6 w = 2 But w =2 causes the denominator of the fractions to become zero therefore it cannot be a solution. (There is no solution)
The Cartesian Coordinate System The Cartesian Coordinate System (Rectangular Coordinate System) is the perpendicular crossing of two number lines (axis), one horizontal (x) and one vertical (y). Using the Cartesian Coordinate System, we can assign names to points on a plane. The point where the two axes cross is called the origin. Origin We name points on the Cartesian Plane by using the ordered pair (x,y), where x (abscissa) is the position along the horizontal (x) axis and y (ordinate) is the position along the vertical (y) axis. (x,y) is called an ordered pair because, in this case, order matters.
The x- axis and the y-axis divide the Cartesian Plane into 4 sections called quadrants (QI, QII, QIII, QIV) Points on either axis are not in any quadrant. The graph of an equation is the set of points that are solutions to the equation 3x + 2y = 12 (4,0) is a solution (0,6) is a solution (1,1) is not a solution
Graphs of the Linear Equation The graph of an equation of two variables is the set of all points that are solutions to the equation. (Obviously, it is impossible to plot all the points that are solutions to equation. We determine the type of graph that is associated with the specific type of equation and plot enough points to position the graph). The Linear Equation An equation of the form Ax + By + C = 0 is called a linear (first degree) equation. The graph of a linear equation is a straight line. To graph a linear equation 1. Find three points that solve the equation (pick 3 values for x and evaluate the corresponding values for y). (Note: Technically you can draw a line using only two points, but we use three for accuracy in our rough sketches) 2. Plot the3 points on a Cartesian Plane. 3. Draw a straight line through the 3 points. (If you cannot draw a straight line between the three points, you probably made a mistake in step 1) Examples: Graph y = x 2 x y = x 2 Point -1 y = 1 2 = -3 (-1, -3) 0 y = 0 2 = -2 (0, -2) 2 y = 2 2 = 0 (2, 0) Plot the points
Draw line through the points Graph y = -2x + 4 x y = -2x + 4 Point 0 y = -2(0) + 4 = 4 (0,4) 1 y = -2(1) + 4 = 2 (1,2) 2 y = 2(2) + 4 = 0 (2,0)
Special Lines A linear equation with B = 0, Ax + C = 0 is represented by a vertical line through the point Example: Graph 2x + 4 = 0 2x + 4 = 0 2x = -4 x = -2 A linear equation with A = 0, By + C = 0 represents a horizontal line through the point Example: Graph 3y 12 = 0 3y = 12 y = 4
Intercept of a Line All straight lines cross at least one of the coordinate axes. The points where the line crosses the axis is called the intercept. The x-intercept is the point where line crosses the x-axis. To find the x-intercept of a line, set y = 0 and solve the resulting equation for x. (x,0) The y-intercept is the point where the line crosses the y-axis. To find the y- intercept of a line, set x = 0 and solve the resulting equation. (0,y) Example: Find the x-intercept and the y-intercept of 3x + 2y = 6 x-intercept y = 0 3x + 2(0) = 6 3x = 6 x = 2 (2,0) y-intercept x = 0 3(0) + 2y = 6 2y = 6 y = 3 (0,3)
Special Cases Vertical Lines (x = K) have only a x-intercept Example: x = 2 Horizontal Lines (y = K) have only a y-intercept Example: y = 3
Slope of a Line We generally use the term slope to refer to the steepness of a line. We also use this term when we discuss the steepness of such physical objects as the slope of a roof or a ski-slope. In algebra, we define the slope of a line as the ratio of the change of Vertical distance (rise) to the change in Horizontal distance (run) between two points rise y 2 y 1 slope = m = =, where (x1,y 1 )and (x 2,y 2 ) run x 2 x 1 are two points on the line. Examples: Find the slope of the line through the points (1,2) and (2,4) 4 2 2 m = = = 2 2 1 1
Find the slope of the line through (0,0) and (2,-4) 4 0 4 m = = = 2 2 0 2 Properties of Slope