PRE-ALGEBRA SUMMARY WHOLE NUMBERS

PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in a number depends on its position or place. 7,,89 Ones Tens Hundreds Thousands Ten thousands Hundred thousands Millions The value of a number is the sum of each digit multiplied by its place value.,4 = (,000) + ( 00) + (4 0) + ( ) Addition of Whole Numbers Addends The numbers that are being added. Sum The result of the addition. + 8 Addends Sum The Properties of Addition The Commutative Property The order in which you add two whole numbers does not affect the sum. The Associative Property The way in which you group whole numbers in addition does not affect the final sum. The Additive Identity The sum of 0 and any whole number is just that whole number. Measuring Perimeter The perimeter is the total distance around the outside edge of a shape. The perimeter of a rectangle is P = L + W + L + W (which can be written as P = L + W). ft + 4 = 4 + ( + 7) + 8 = + (7 + 8) 6 ft 6 ft 6 + 0 = 0 + 6 = 6 ft P = L + W + L + W = 6 ft + ft + 6 ft + ft = 6 ft Subtraction of Whole Numbers Minuend The number we are subtracting from. Subtrahend The number that is being subtracted. Difference The result of the subtraction. Minuend + 8 Subtrahend Difference 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY WHOLE NUMBERS Rounding, Estimation, and Ordering of Whole Numbers Step To round a whole number to a certain decimal place, look at the digit to the right of that place. Step a. If that digit is or more, that digit and all digits to the right become 0. The digit in the place you are rounding to is increased by. b. If that digit is less than, that digit and all digits to the right become 0. The digit in the place your are rounding to remains the same. Order on the Whole Numbers For the numbers a and b, we can write. a < b (read a is less than b ) when a is to the left of b on the number line. To the nearest hundred, 4,78 is rounded to 4,600. To the nearest thousand, 7, is rounded to 7,000. 8 <. a > b (read a is greater than b ) when a is to the right of b on the number line. > Multiplication of Whole Numbers 8 9 0 4 Factors The numbers being multiplied. Product The result of the multiplication. The Properties of Multiplication The Commutative Property Multiplication, like addition, is a commutative operation. The order in which you multiply two whole numbers does not affect the product. The Distributive Property To multiply a factor by a sum of numbers, multiply the factor by each number inside the parentheses. Then add the products. Multiplicative Property of Zero The product of zero and any number is zero. Multiplicative Identity Property The product of one and any number is that number. The Associative Property Multiplication is an associative operation. The way in which you group numbers in multiplication does not affect the final product. 7 9 = 6 Product Factors 7 9 = 9 7 ( + 7) = ( ) + ( 7) 0 = 0 = 0 = = ( ) 6 = ( 6) Finding the Area of a Rectangle The area of a rectangle is found using the formula A = L W. 6 ft ft A = L W = 6 ft ft = ft 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY WHOLE NUMBERS Division of Whole Numbers Divisor The number we are dividing by. Dividend The number being divided. Quotient The result of the division. Remainder The number left over after the division. Divisor 7 8 Quotient Dividend Remainder Dividend = divisor quotient + remainder 8 = 7 + The Role of 0 in Division Zero divided by any whole number (except 0) is 0. 0 7 = 0 Division by 0 is undefined. 7 0 = undefined. Exponents and Whole Numbers Using Exponents Base The number that is raised to a power. Exponent The exponent is written to the right and above the base. The exponent tells the number of times the base is to be used as a factor. Base Exponent = = Three Factors This is read to the third power or cubed Grouping Symbols and the Order of Operations The Order of Operations Mixed operations in an expression should be done in the following order: Step Do any operations inside parenthesis. Step Evaluate any exponents. Step Do all multiplication and division in order from left to right. Step 4 Do all addition and subtraction in order from left to right. Remember Please Excuse My Dear Aunt Sally. 4 ( + ) 7 = 4 7 = 4 7 = 00 7 = 9 An Introduction to Equations An expression is a number or a meaningful collection of operations (+,,, ) and numbers. An equation is two expressions connected by an equal sign. An equation can be true or false. 9 + 9 = 9 + 4 = 8 is true 4 = is false 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY INTEGERS AND INTRODUCTION TO ALGEBRA Introduction to Integers Positive Integers Integers used to name whole numbers to the right of the origin on the number line. Negative Integers Integers used to name the opposites of whole numbers. Negatives are found to the left of the origin on the number line. Integers Whole numbers and their opposites. The integers are {, -, -, -, 0,,,, } Absolute Value The distance (on the number line) between the point named by a signed number and the origin. The absolute value of x is written x. The origin - - - 0 Negative Integers 7 = 7 0 = 0 Positive Integers Addition of Integers Adding Integers. If two integers have the same sign, add their absolute values. Give the result the sign of the original integers.. If two integers have different signs, subtract their absolute values, the smaller from the larger. Give the result the sign of the integer with the larger absolute value. 9 + 7 = 6 (-9) + (-7) = -6 + (-0) = (-) + 9 = - Subtraction of Integers Subtracting Integers. Rewrite the subtraction problem as an addition problem by: a. Changing the subtraction symbol to an addition symbol b. Replacing the integer being subtracted with its opposite. Add the resulting integers as before. 6 8 = 6 + (-8) = 8 8 = 8 + (-) = -7-9 (-7) = -9 + 7 = - Multiplication of Integers Multiplying Integers Multiply the absolute values of the two integers.. If the integers have different signs, the product is negative.. If the integers have the same sign, the product is positive. (-7) = - (-0)(9) = -90 8 7 = 6 (-9)(-8) = 7 (-) = (-)(-) = 4 - = -( ) = -4 00 McGraw-Hill Companies 4

PRE-ALGEBRA SUMMARY INTEGERS AND INTRODUCTION TO ALGEBRA Division of Integers Dividing Integers Divide the absolute values of the two integers.. If the integers have different signs, the quotient is negative.. If the integers have the same sign, the quotient is positive 8 4 8 9 Introduction to Algebra: Variables and Expressions Multiplication x y (x)(y) xy These all mean the product of x and y or x times y The product of m and n is mn. The product of and the sum of a and b is (a + b) Evaluating Algebraic Expressions Evaluating Algebraic Expressions To evaluate an algebraic expression:. Replace each variable or letter with its value.. Do the necessary arithmetic, following the rules for order of operations. Evaluate x + y if x = and y = -. x + y = () + (-) = 0 + (-6) = 4 Simplifying Algebraic Expressions Term A number or the product of a number and one or more variables xy is a term Combining Like Terms To combine like terms:. Add or subtract the coefficients (the numbers multiplying the variables). Attach the common variable. x + x = 7x 8a a = a Introduction to Linear Equations Equation A statement that two expressions are equal Solution A value for a variable that makes an equation a true statement. x = is an equation 4 is a solution for the above equation because (4) = The Addition Property of Equality Equivalent Equations Equations that have exactly the same solutions. x = and x = 4 are equivalent equations The Addition Property of Equality If a = b, then a + c = b + c If x = = 7, then x + = 7 + 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY FRACTIONS AND EQUATIONS Introduction to Fractions Fraction Fractions name a number of equal parts of a unit or whole. A fraction is written in the form b a, in which a is an integer and b is a natural number. is a fraction. 8 Numerator The number of parts of the whole that are being considered. Denominator The number of equal parts into which the whole is divided. 8 Numerator Denominator Proper Fraction A fraction whose numerator is less than its denominator. It names a number less than. Improper Fraction A fraction whose numerator is greater than or equal to its denominator. It names a number greater than or equal to. and are proper fractions. 7, 0 and 8 8 are improper fractions. Prime Numbers and Factorization Prime Number Any natural number greater than that has exactly two factors, and itself. Composite Number Any natural number greater than that is not prime. Prime Factorization To find the prime factorization of a number, divide the number by a series of primes until the final quotient is a prime number. The prime factors include each prime divisor and the final quotient. 7,, 9, and 7 are prime numbers. 8,, 4, and 6 are composite numbers. 60 = 7 Greatest Common Factor (GCF) The GCF is the largest number that is a factor of each of a group of numbers. To Find the GCF Step Write the prime factorization for each of the numbers in the group. Step Locate the prime factors that are common to all the numbers. Step The greatest common factor (GCF) will be the product of all the common prime factors. If there are no common prime factors, the GCF is. To find the GCF of 4, 0, and 6: The GCF is = 6 00 McGraw-Hill Companies 6

PRE-ALGEBRA SUMMARY FRACTIONS AND EQUATIONS Equivalent Fractions The Fundamental Principal of Fractions For the fraction b a, and any nonzero number c, a a c b b c In words: We can divide the numerator and denominator of a fraction by the same nonzero number. This is used to simply (or reduce) a fraction. Equivalent Fractions The fundamental principal can also be written as a a c c 0 b b c This is used to build up an equivalent fraction. 8 8 4 4 8 and are equivalent fractions. 0 Multiplication and Division of Fractions To Multiply Two Fractions. Multiply numerator by numerator. This gives the numerator of the product.. Multiply denominator by denominator. This gives the denominator of the product.. Simplify the resulting fraction if possible. In simplifying fractions, it is usually easiest to divide by any common factors in the numerator and denominator before multiplying. To Divide Two Fractions Replace the divisor by its reciprocal and multiply. 8 7 8 7 6 9 7 0 4 9 0 7 4 6 8 Linear Equations in One Variable Solving Linear Equations The steps for solving a linear equation are as follows:. Use the distributive property to remove any grouping symbols. Then simply by combining like terms.. Add or subtract the same term on both sides of the equation until the variable term is on one side and a number is on the other.. Multiply or divide both sides of the equation by the same nonzero number so that the variable is alone on one side of the equation 4. Check the solution in the original equation. Solve: (x ) + 4x = x + 4 x 6 + 4x = x + 4 7x 6 = x + 4 + 6 = + 6 7x = x + 0 -x = -x 4x = 0 4x 0 4 4 x = 00 McGraw-Hill Companies 7

PRE-ALGEBRA SUMMARY APPLICATIONS OF FRACTIONS AND EQUATIONS Addition and Subtraction of Fractions To Add (Subtract) Like Fractions. Add (subtract) the numerators.. Place the sum (difference) over the common denominator.. Simplify the resulting fraction if necessary. 8 7 8 8 Least Common Multiple (LCM) The LCM is the smallest number that is a multiple of each of a group of numbers. To Find the LCD of a Group of Fractions. Write the prime factorization for each of the denominators.. Find all the prime factors that appear in any one of the prime factorizations.. Form the product of those prime factors, using each factor the greatest number of times it occurs in any one factorization. To find the LCD of fractions with denominators 4, 6, and : 4 = 6 = = The LCD =, or 60 To Add or Subtract Unlike Fractions. Find the LCD of the fractions. 7. Change each fraction to an equivalent fraction with the LCD as a common 4 0 0 denominator. 9. Add (subtract) the resulting like fractions as before. 0 Operations on Mixed Numbers 4 0 Mixed Number The sum of a whole number and a proper fraction To Change an Improper Fraction into a Mixed Number. Divide the numerator by the denominator. The quotient is the wholenumber portion of the mixed number.. If there is a remainder, write the remainder over the original denominator. This gives the fractional portion of the mixed number. and 7 are mixed numbers. 8 Note that means 4. 4 Quotient 0 Remainder To Change a Mixed Number to an Improper Fraction. Multiply the denominator of the fraction by the whole-number portion of the mixed number.. Add the numerator of the fraction to that product.. Write the sum over the original denominator to form the improper fraction. Denominator Whole Number Numerator (4) 4 4 4 Denominator 00 McGraw-Hill Companies 8

PRE-ALGEBRA SUMMARY APPLICATIONS OF FRACTIONS AND EQUATIONS Addition and Subtraction of Fractions Multiplying or Dividing Mixed Numbers 4 Convert any mixed or whole numbers to improper fractions. 0 6 64 6 Then multiply or divide the fractions as before. To Add or Subtract Mixed Numbers. Rewrite as improper fractions.. Add or subtract the fractions.. Rewrite the results as a mixed number if required. 4 4 4 4 7 4 4 Equations Containing Fractions To Solve an equation containing one or more fractions. Find the LCD of the denominators.. Multiply every term by the LCD.. Solve the resulting equation as before. x To solve ;. The LCD is 0. x. 0( ) 0( ). x + 0 = x = - x Applications of Linear Equations in One Variable To Use an Equation to Solve a Word Problem. Read the problem carefully to decide what you are asked to find.. Choose a letter to represent one of the unknowns. Then represent all other unknowns with expressions using that same letter.. Translate the problem to algebra to form an equation. 4. Solve the equation and answer the original question.. Check your solution by returning to the original problem. Consecutive Integers If x is an integer, then x + is the next consecutive integer, x + is the next, and so on. If 0 is an integer, 0 + = is the next consecutive integer. Complex Fractions A complex fraction has a fraction in its numerator or denominator (or both) To simplify a complex fraction, multiply the numerator and denominator by the LCD of the fractions within the complex fraction. is a complex fraction. 6 6 6 6 6 4 00 McGraw-Hill Companies 9

PRE-ALGEBRA SUMMARY DECIMALS Introduction to Decimals, Place Value, and Rounding Decimal Fraction A fraction whose denominator is a power of 0. We call decimal fractions decimals. 7 47 and are decimal fractions. 0 00 Decimal Place Each position for a digit to the right of the decimal point. Each decimal place has a place value that is one-tenth the value of the place to its left..46 Ten thousandths Thousandths Hundredths Tenths Reading and Writing Decimals in Words. Read the digits to the left of the decimal point as a whole number.. Read the decimal point as the word and.. Read the digits to the right of the decimal point as a whole number followed by the place value of the rightmost digit. Rounding Decimals. Find the place to which the decimal is to be rounded.. If the next digit to the right is or more, increase the digit in the place you are rounding by. Discard any remaining digits to the right.. If the next digit to the right is less than, just discard that digit and any remaining digits to the right. Hundredths 8. is read eight and fifteen hundredths. To round.87 to the nearest tenth:.87 is rounded to.9 To round.44 to the nearest thousandth:.44 is rounded to.4 Addition and Subtraction of Decimals To Add or Subtract Decimals. Write the numbers being added (or subtracted) in column form with their decimal points in a vertical line. You may have to place zeros to the right of the existing digits.. Add (or subtract) just as you would with whole numbers.. Place the decimal point of the sum (or difference) in line with the decimal points. To subtract.87 from 8.: 8.00.87.6 Multiplication of Decimals To Multiply Decimals. Multiply the decimals as though they were whole numbers.. Add the number of decimal places in the factors.. Place the decimal point in the product so that the number of decimal places in the product is the sum of the number of decimal places in the factors. Multiplying by Powers of 0 Move the decimal point to the right the same number of places as there are zeros in the power of 0. To multiply.8 0.04:.8 Two places x 0.04 Three places 4 40 0.8 Five places.7 0 =.7 0.67,000 = 67 00 McGraw-Hill Companies 0

PRE-ALGEBRA SUMMARY DECIMALS Division of Decimals To Divide by a Decimal. Move the decimal point to the right, making the divisor a whole number.. Move the decimal point in the dividend to the right the same number of places. Add zeros if necessary.. Place the decimal point in the quotient directly above the decimal point in the dividend. 4. Divide as you would with whole numbers To Divide by a Power of 0 Move the decimal point to the left the same number of places as there are zeros in the power of 0. To divide 6. by., move the decimal points:. 6. 6. 0.8 0 = ^.8=.8 Fractions and Decimals To Convert a Common Fraction to a Decimal. Divide the numerator of the common fraction by its denominator.. The quotient is the decimal equivalent of the common fraction. To Convert a Terminating Decimal Less Than to a Common Fraction. Write the digits of the decimal without the decimal point. This will be the numerator of the common fraction.. The denominator of the fraction is a followed by as many zeros as there are places in the decimal. To convert to a decimal: 0..0 0 0 To convert 0.7 to a common fraction: 7 0.7 000 40 Equations Containing Decimals To solve an equation that contains decimals, use the same procedure used for solving other linear equations..x +.9x =.7x..x =.7x..4x = -.. x =.4 x =. Square Roots and the Pythagorean Theorem The square root of a number is a value that, when squared, gives us that number. The length of the three sides of a right triangle will form a perfect triple. 4 + 4 = The Pythagorean theorem is usually written as c = a + b a b c 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY RATIO, RATE, AND PROPORTION Ratios Ratio A means of comparing two numbers or quantities. A ratio can be written as a fraction. 4 can be thought of a the ratio of 4 to 7. 7 Rates Rate A fraction involving two denominate numbers with different units. Unit Price The cost per unit. 0home runs home run 0games games $ rolls $0.40per roll Proportions Proportion A statement that two ratios or rates are equal. 6 is a proportion that reads, three is 0 to five as six is to ten. The Proportion Rule If a c 6, then a d = b c If, then 6 = 0 b d 0 To Solve a Proportion. Use the proportion rule to write the equivalent equation a d = b c.. Divide both terms of the equation by the coefficient of the variable.. Use the value found to replace the unknown in the original proportion. Check that the ratios or rates are proportional. To solve: x 6 0 0x 6 0x 80 0 x 0 x 4 80 0 Similar Triangles and Proportions A triangle in which two of the sides are perpendicular is called a right triangle. is a right triangle. Two right triangles are similar if their corresponding sides are proportional. 6 are proportional. 0 If we know that two triangles are similar, we can use a proportion to find the length of a missing side. 8 8 x x 8 x 4 x x 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY RATIO, RATE, AND PROPORTION Linear Measurement and Conversion The English system of measurement is in common use in the United States. English Units of Measurement and Equivalents Length Weight foot (ft) = inches (in.) pound (lb) =6 ounces (oz) yard (yd) = ft ton =,000 lb mile (mi) =,80 ft Capacity pint (pt) = 6 fluid ounces (fl oz) quart (qt) = pt gallon (gal) = 4 qt Units fractions A fraction whose value is. Units fractions can be used to convert units. To Add Like Units of Length. Arrange the numbers so that the like units are in the same column.. Add in each column.. Simply if necessary. To Subtract Like Units of Length. Arrange the numbers so that the like units are in the same column.. Subtract in each column. You may have to borrow from the larger unit at this point.. Simplify if necessary. To Multiply or Divide Units by Abstract Numbers. Multiply or divide each part of the measurement by the abstract number.. Simplify if necessary. in. ft and are unit fractions. ft yd To add 4 ft 7 in. and ft 0 in.: To subtract: 4 ft 7 in. + ft 0 in. 9 ft 7 in. = 0 ft in. 4 ft 7 in. ft 9 in. Borrow and rename: ft 9 in. ft 9 in. ft 0 in. ( yd ft) = 6 yd 4 ft, or 7 yd ft Metric Units of Length The metric system of measurement is used throughout most of the world. Common metric units of length are the meter (m), centimeter (cm), millimeter (mm), and kilometer (km). Basic Metric Prefixes milli* means,000 centi* means 00 deci means 0 kilo* means,000 hecto means 00 deka means 0 * These are the most commonly used and should be memorized. 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY PERCENT Percents, Decimals, and Fractions Percent Another way of naming parts of a whole. Percent means per hundred.. To convert a percent to a fraction, divide by 00.. To convert a percent to a decimal, remove the percent symbol and move the decimal point two places to the left.. To convert a decimal to a percent, move the decimal point two places to the right and attach the percent symbol. 4. To convert a fraction to a percent, write the decimal equivalent of the fraction and then change that decimal to a percent. Fractions and decimals are other ways of naming parts of a whole. % 0. 00 7% 7 00 7% = 0.7 0.8 = 8% 0.60 60% Solving Percent Problems Using Proportions Every percent problem has the following three parts:. The base. This is the whole in the problem. It is the standard used for comparison. Label the base B.. The amount This is the part of the whole being compared to the base. Label the amount A.. The rate. This is the ratio of the amount to the base. The rate is written as r a percent. Label the rate or R. 00 4 is 0% of 0 A R B Using the Percent Proportion The percent proportion is A B r 00 To solve a percent problem using this proportion:. Substitute the two known values into the proportion.. Solve the proportion to find the unknown value. What is 4% of 00? A 00 4 00 00A = 7,00 A = 7 00 McGraw-Hill Companies 4

PRE-ALGEBRA SUMMARY PERCENT Solving Percent Applications Using Equations Translating a question into an equation: What is r% of B? r A B 00 What is % of 00? A 00 00 A = A = 69 Sales tax, commissions, and discounts are all rates. Applications can be solved by the percent proportion or by translating the question into an equation. If a % commission is $,000, what is the total?,000 B 00 B = 00,000 B = $,000 Applications: Simple and Compound Interest Simple interest is a rate. Compound interest is interest on an amount that includes previous interest. 00 McGraw-Hill Companies

PRE-ALGEBRA SUMMARY GEOMETRY Lines and Angles Line A series of points that goes on forever. Line segment A piece of a line that has two endpoints. Angle A geometric figure consisting of two line segments that share a common endpoint. Perpendicular lines Lines are perpendicular if they intersect to form four equal angles. A A O B B C D Parallel lines Lines are parallel if they never intersect. Right angles have a measure of 90 o. Acute angles have a measure less than 90 o. Obtuse angles have a measure between 90 o and 80 o. O A B C E F 00 McGraw-Hill Companies 6

PRE-ALGEBRA SUMMARY GEOMETRY Perimeter and Circumference A polygon is a closed figure with three or more sides. Each side is a line segment. Triangle Square The perimeter is the sum of the lengths of its sides: Rectangle Circumference Circle P = a + b + c P = 4s P = L + W C = πr Find the perimeter of the figure: P cm.6 cm = L + W = (.6 cm) + ( cm) =. cm + 0 cm =. cm Area and Volume The area of an object is the number of unit squares needed to cover its surface. Square A = s Rectangle A = L W Triangle A b h Parallelogram A = b h Circle A = πr Some common volume formulas: Cube V = s Rectangular solid V = L W H Sphere V = 4 r Cylinder V = πr h Find the area of the figure: A (6 in.)(.4 in.) = ( in.)(.4 in) = 7. in. Find the area of the figure:.4 in. 6 in. h =4 cm r = cm V = πr h = π( cm) (4 cm) = 6π cm =. cm 00 McGraw-Hill Companies 7

PRE-ALGEBRA SUMMARY GRAPHING AND INTRODUCTION TO STATISTICS Tables and Graphs of Data A table is a display of information in parallel columns or rows. A graph is a diagram that relates two different pieces of information. One of the most common graphs is the bar graph. Amount 4 Day In line graphs, one of the axes is usually related to time. The Rectangular Coordinate System The Rectangular Coordinate system The rectangular coordinate system is a system formed by two perpendicular axes that intersect at a point called the origin. The horizontal line is called the x-axis. The vertical line is called the y-axis. Graphing Points from Ordered Pairs The coordinates of an ordered pair allow you to associate a point in the plain with every ordered pair. To graph a point in the plane, Step Start at the origin. Step Move right or left according to the value of the x-coordinate: to the right if x is positive or to the left if x is negative. Step Then move up or down according to the value of the y-coordinate: up if y is positive and down if y is negative. To graph the point corresponding to (, ) y (, ) units x units Linear Equations in Two Variables Solutions of Linear Equations A pair of values that satisfies the equation. Solutions for linear equations in two variables are written as ordered pairs. An ordered pair has the form If x y = 0, (6, ) is a solution for the equation, because substituting 6 for x and for y gives a true statement. (x, y) x-coordinate y-coordinate Linear Equations in Two Variables 00 McGraw-Hill Companies 8

PRE-ALGEBRA SUMMARY GRAPHING AND INTRODUCTION TO STATISTICS Linear Equation An equation that can be written in the form Ax + By = C in which A and B are not both 0. x y = 4 is a linear equation. Graphing Linear Equations Step Find at least three solutions for the equation and put your results in tabular form. Step Graph the solutions found in step. Step Draw a straight line through the points determined in step to form the graph of the equation. x - y = 6 (6, 0) (, -) (0, -6) Mean, Median, and Mode Finding the Mean To find the mean for a group of numbers follow these two steps: Step Add all the numbers in the group. Step Divide that sum by the number of items in the group. Given the numbers 4, 8, 7, 4 + 8 + 7 + = Mean = 4 Finding the Median The median is the number for which there are as many instances that are to the right of that number as there are instances to the left of it. To find the median follow these steps: Step Rewrite the numbers in order from smallest to largest. Step Count from both ends to find the number in the middle. Step If there are two numbers in the middle, add them together and find their mean. Given the numbers 9,,,, 7, Rewrite them:,,, 7, 9, The middle numbers are and 7 7 6 Median = 6 Finding the Mode The mode is the number that occurs most frequently in a set of numbers. Given the numbers,,,,,, 7, 7, 9,. The mode is. 00 McGraw-Hill Companies 9

PRE-ALGEBRA SUMMARY POLYNOMIALS Properties of Exponents. a m a n = a m+n. (a m ) n = a mn. (ab) m =a m b m m a mn 4. a ( a 0) n a m a m a. ( ) ( b 0) b m b x x = x (x ) = x 6 (4x) =4 x =64x x x 7 x ( ) y x x y Introduction to Polynomials Term A term is a number or the product of a number and variables. Polynomial A polynomial is an algebraic expression made up of terms in which the exponents are whole numbers. These terms are connected by plus or minus signs. Each sign (+ or -) is attached to the term following that sign. Coefficient In each term of a polynomial, the number in front of the variable is called the numerical coefficient or, more simply, the coefficient of that term. Types of Polynomials A polynomial can be classified according to the number of terms it has. A monomial has one term. A binomial has two terms. A trinomial has three terms. 4x x + x is a polynomial. The terms of 4x x + x are 4x, -x, and x. The coefficients of 4x x are 4 and -. x is a monomial. x 7x is a binomial. x x + is a trinomial. Addition and Subtraction of Polynomials Removing Signs of Grouping. If a plus sign (+) or no sign at all appears in front of parentheses, just remove the parentheses. No other changes are necessary.. If a minus sign (-) appears in front of parentheses, the parentheses can be removed by changing the sign of each term inside the parentheses. Adding Polynomials Remove the signs of grouping. Then collect and combine any like terms. Subtracting Polynomials Remove the signs of grouping by changing the sign of each term in the polynomial being subtracted. Then combine any like terms. x + (x ) = x + x x (x 4) = x x + 4 (x +) + (x ) = x + + x = x (x + x) (x + x ) = x + x x -x + Signs change = x x + x x + = x x + 00 McGraw-Hill Companies 0

PRE-ALGEBRA SUMMARY POLYNOMIALS Multiplying Polynomials To Multiply a Polynomial by a Monomial Multiply each term of the polynomial by the monomial and simplify the results. x(x + ) = x x + x = 6x + 9x To Multiply a Binomial by a Binomial Use the FOIL method: F O I L (a + b)(c + d) = a c + a d + b c + b d To Multiply a Polynomial by a Polynomial Arrange the polynomials vertically. Multiply each term of the upper polynomial by each term of the lower polynomial and add the results. (x )(x + ) = 6x + 0x 9x F O I L = 6x + x x x + x -x + 9x x 6x +0x x 9x +9x Introduction to Factoring Polynomials Common Monomial Factor A common monomial factor is a single term that is a factor of every term of the polynomial. The greatest common factor (GCF) of a polynomial is the common monomial factor that has the largest possible numerical coefficient and the largest possible exponents. Factoring a Monomial from a Polynomial. Determine the GCF for all terms.. Factor the GCF from each term and then apply the distributive property in the form ab + ac = a(b + c) The greatest common factor. Mentally check by multiplication 4x is the greatest common factor of 8x 4 x + 6x. 8x 4 x + 6x = 4x (x x +4) 00 McGraw-Hill Companies