MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.


 Mitchell Lee
 8 months ago
 Views:
Transcription
1 MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division. All rights reserved. No part of this book may be reproduced in any form without written permission from the Certified General Accountant Association of Canada and the UBC Real Estate Division. Published by the UBC Real Estate Division. Printed in Vancouver, Canada. RMATH0
2
3 MATH REVIEW KIT Table of Contents Page SECTION : THE NUMBER SYSTEM ARITHMETIC OF SIGNED (NEGATIVE OR POSITIVE) NUMBERS INFINITY PROBLEMS...  SECTION : SIMPLE ALGEBRAIC OPERATIONS PROPERTIES OF ADDITION PROPERTIES OF MULTIPLICATION DISTRIBUTIVE LAW OF ADDITION AND MULTIPLICATION EQUALITY AXIOMS PROBLEMS SECTION : COMPOSITE ALGEBRAIC OPERATIONS ALGEBRAIC EXPRESSIONS GROUPING SYMBOLS CORRECT USE OF PARENTHESES PROBLEMS...  SECTION 4: ALGEBRA OF FRACTIONS THE LOWEST FORM OF FRACTIONS CONVERTING FRACTIONS TO A COMMON DENOMINATOR ALGEBRA OF FRACTIONS PERCENTAGES ROUNDING OFF A DECIMAL COMMON MISTAKES IN THE USE OF FRACTIONS PROBLEMS SECTION : ALGEBRA OF EXPONENTS MEANING OF EXPONENT ALGEBRA OF EXPONENTS COMPOUND INTEREST PROBLEMS SECTION 6: LINEAR FUNCTIONS AND INTERPOLATION LINEAR INTERPOLATION PROBLEMS SECTION 7: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES LINEAR EQUATIONS IN ONE VARIABLE PROBLEMS SECTION 8: IDENTITIES AND FACTORIZATION THE IDENTITIES FACTORIZATION PROBLEMS SOLUTIONS... Solutions
4 SECTION THE NUMBER SYSTEM. ARITHMETIC OF SIGNED (NEGATIVE OR POSITIVE) NUMBERS i) Negative of a negative is its absolute value, i.e., () =. This is because () + = 0. To add two numbers of the same sign, we add the absolute values and assign the sign of the two numbers. Example : + 8 =, + ( 9) = 4 To add two numbers of different signs, take the difference of the absolute values and attach the sign of the number with the greater absolute value. Example : + ( 8) =, + 7 = 8, 9 = 7, and + 9 = 8. i To find the product or quotient of two numbers, multiply or divide the absolute values and attach a positive sign if the two numbers have the same sign; otherwise, attach a negative sign. Example : = 7, ( ) ( ) =, ( 0) = 00, ( 8) 6 = 48, & & =, =, =. & &. INFINITY Definition (Infinity). Infinity denoted as is a number larger than any given number. In fact there is no number that would satisfy definition 0. The idea of infinity is to provide some kind of conceptual bounds to the number line, precisely and +, the negative and positive infinities. Mathematical manipulations involving infinite numbers are different from that with finite numbers. The following propositions depict some characteristics: Proposition : There are as many even numbers as there are natural numbers. Consider the natural and even numbers listed as follows:
5 Math Review Kit: Section If we associate to, to 4, to 6, etc., then for every natural number, we are able to produce an even number and vice versa. Proposition : There are infinitely many rational numbers between any two rational numbers. To illustrate this, take two rationals 0 and. Referring to Figure, we can generate numbers, /, /4, /8, /6,... breaking up the lengths to half and moving to the left. There is an infinite number of numbers between 0 and. Figure. PROBLEMS. Indicate whether the following numbers are rational or irrational. &8 B 6,, ,,,, , e, Solve the following: i) + ( 6) 6 + ( 9) i + ( 8) iv) 7 ( ) v) 7 (+6) vi) (8)( ) v + (0) vi ( )( 7) ix) (&6) & (&7) (&)(&) Note that: [(A)(B)] = [A B] = [AB] 
6 SECTION SIMPLE ALGEBRAIC OPERATIONS Manipulation of numbers comes within the realm of arithmetic. When we state a general rule of arithmetic, we have progressed to algebra. An arbitrary number may be denoted by a, b, c, d or x, y, z, etc. In this section we study the properties of basic algebraic operations of addition and multiplication and discuss the algebraic structure of numbers.. PROPERTIES OF ADDITION The operation of addition of two arbitrary numbers a and b is the number denoted by a + b. Example : i) If a = 9, b = then a + b = 9 + = 4. Adding to the number a yields + a. Definition (Identity of addition). The number 0 is called the identity of addition with the properties: a + 0 = a () 0 + a = a () Definition (Additive inverse). The additive inverse of a number a is a number b such that a + b = 0 () and b + a = 0 (4) The additive inverse of a is a. Example : Find the additive inverses of and . Since + ( ) = 0 and ( ) + = 0, the additive inverse of is. Similarly, the additive inverse of is. Thus, the process of subtraction can be interpreted in terms of addition. 6 = 4 could be described as the addition: 6 + ( ) = 4. Commutative Law of Addition. If a and b are any two numbers then a + b = b + a () Thus: + = + = 7 and + = + ( ) =. 
7 Math Review Kit: Section Associative Law of Addition. If a, b and c are any three numbers, then: (a + b) + c = a + (b + c) (6) The parentheses in the above equation indicate the order in which the operation (addition) is performed. Thus: + ( + 8) = + = and ( + ) + 8 = =. Example : Consider the following scheme of addition of large numbers: This conventional way of addition involves a combination of commutative and associative laws of addition which can be paraphrased as: = (0 + ) + (80 + ) + (0 + 9) = ( ) + ( + + 9) = = 47.. PROPERTIES OF MULTIPLICATION The operation of multiplication of two arbitrary numbers a and b is a number denoted by a b (read as a times b ); a b can be written as ab. Example 4: i) If a =, b = Then ab = = 6. If a =, b = 6 then ab = ( 6) =. i times a is a. Definition (Identity of Multiplication). The number is called the identity of multiplication with the properties: a = a (7) a = a (8) 
8 Simple Algebraic Operations Definition 4 (Multiplicative Inverse). The multiplicative inverse of a number a is a number b such that: ab = (9) ba = (0) The multiplicative inverse of a is /a, provided a =/ 0 (a is not zero). Example : Find the multiplicative inverses of 8 and. As 8 /8 = and /8 8 =, /8 is the multiplicative inverse of 8. Similarly, / is the multiplicative inverse of. Thus division of a by b (b =/ 0) can be identified with the multiplication of a with the multiplicative inverse of b, i.e., a b the same as a /b. Proposition : Every real number has a unique additive inverse but only a nonzero number has a (unique) multiplicative inverse. The multiplicative inverse of 0, which is /0 =, is not defined. Commutative Law of Multiplication. If a and b are any two numbers then: ab = ba () Thus: = = 6 ( 9) = ( 9) = 4. Associative Law of Multiplication. If a, b and c are any three numbers, then: a(bc) = (ab)c () Thus: 4 (8 ) = 4 4 = 96 and (4 8) = = 96.. DISTRIBUTIVE LAW OF ADDITION AND MULTIPLICATION Distributive Law. If a, b and c are any three numbers, then: a(b + c) = ab + ac () 
9 Math Review Kit: Section Using the rule (), () can also be written as: (b + c)a = ba + ca (4) Thus: ( + ) = ( ) + ( ) = + 0 =. Example 6: Demonstrate the role of distributive law in the following scheme of multiplication: = (0 + 9) (40 + ) = {(0 + 9) } + {(0 + 9) 40} = {(0 ) + (9 )} + {(0 40) + (9 40)} = = 7.4 EQUALITY AXIOMS When a is equal to b, we write a = b. Following are the selfevident facts (axioms) that the equality relation satisfies: Reflexive: a = a () Symmetric: if a = b, then b = a (6) Transitive: if a = b and b = c, then a = c (7) If a = b, then a +_ c = b +_ c (8) If a = b, then ac = bc (9) If a = b and c = d, then a +_ c = b +_ d (0) If a = b and c = d, then ac = bd () If a + c = b + c, then a = b () If ac = bc and c =/ 0, then a = b (). PROBLEMS. Find the additive inverse of the following numbers: 8, 9, 9, 0, 6,. 4
10 Simple Algebraic Operations. Find the multiplicative inverse of the numbers:,, 9, 9,.. Check whether the following statements are true or false. i) a(b + c) = ab + ac (a +b) = a + b i a(b + ) = ab + b + iv) x(y z) = xy xz v) (y + z w)x = xy + xz xw vi) v (x y)(z w) = xz xw yz + yw (a + b)( c) = a + b ca + cb 
11 SECTION COMPOSITE ALGEBRAIC OPERATIONS In this section, we discuss more complicated algebraic expressions and the use of grouping symbols.. ALGEBRAIC EXPRESSIONS Definition (Constant). A constant is a number or an algebraic symbol that stands for a particular value in a given context. For example, 7,, π, e, are all constants. Sometimes a letter c may denote a constant. Definition (Variable). A variable is an algebraic symbol, say x, that may take on different values from the real number system. Definition (Algebraic Expression). An algebraic expression is created when one or more algebraic operations are performed upon variables and constants. It may consist of one or more terms, separated from each other by the signs + and. For example, x + y + xy z is an algebraic expression consisting of four terms. When the terms do not differ or differ only in their numerical coefficients, they are called like terms. Thus, x and 8x; ab and ab; xyz and 4xyz are pairs of like terms. Abbreviation of Some Algebraic Terms i) The sum a + a is written as a. Similarly, a + a + a = a, and so on. The term a also means a. Addition of like terms is obtained by taking the algebraic sum of the coefficients and multiplying by the term. Thus, x + x = 6x, 8ab ab + ab = (8 + l)ab = 7ab. i The product a a means a and a a a means a and so on. iv) Square root of x, denoted by x, is a number y such that y = x. Similarly a cube root of x, x symbolized as, is a number z such that z = x. For example and  are square roots of 4 because = 4 and () = 4. The two square roots are sometimes written as ± 4. Example : i) Find the value of (x x 6) if x =. () () 6 = = 6 = 
12 Math Review Kit: Section Find the value of (x + y) if x = and y =. ( ) = ( ) = 4 i Find the value of (x & x % ) for x =. ( ) & % = 0 = iv) When multiplying two expressions, the multiplication sign (x) may be dispensed with. The distributive law can be used to simplify the product. Example : Simplify the following: i) (x + ) = x + 6 [Using the distributive law] = x + x + ( x) = x + x [Treating + ( x) as ( x), and using the distributive law] = x + Thus, parentheses preceded by a + sign may be removed without changing the expression. i y (x + y ) = y x y + [Treating (x + y ) as (x + y ), and using the distributive law] = y x Thus, parentheses preceded by a  sign may be removed by changing the sign of each term of the expression within the parentheses.. GROUPING SYMBOLS An algebraic expression may involve grouping symbols: parenthesis ( ), brackets [ ], and braces { }. These are used to indicate the order in which the basic operations of +,,, and " are performed. We now consider the rules of simplifying expressions involving composite algebraic operations. 
13 Composite Algebraic Operations Expressions with no Grouping Symbol Rule : In the order of operations, multiplication and division belong to the same hierarchy followed by the hierarchy of addition and subtraction. When two operations in the same hierarchy appear one after the other, merely proceed from left to right. Example : Simplify: i) 0 = 6 = = 0 = = = + = = Expressions with more than one Grouping Symbol Rule : First solve the innermost parentheses, then brackets and then outermost braces. Example 4: Simplify: i) [ (0 9)] 6 4 = [ ()] 6 4 = [] 6 4 = 4 = 
14 Math Review Kit: Section x [4x + x(x )] = x [4x + x x] = x [x + x ] = x x x = x x i {8 + [ 9( + 6)] } = {40 + [ 9()] } = {40 + [ 8] } = {40 + [] } = {40 0 } = {40 } = {8} = 76 iv) a 4{a [b + (a ) (b )]} = a 4{a [b + a 6b + 0]} = a 4{a [a 4b + ]} = a 4{a a + b } = a 4{4a + b } = a + 6a 48b + 60 = 8a 48b
15 Composite Algebraic Operations. CORRECT USE OF PARENTHESES The following examples depict some common mistakes in using or not using parentheses. Example : i) From x subtract less than a number y. Answer: x y (incorrect) The correct answer is x (y ). Find the area of a rectangle with sides a and a units. Answer: (a) (a) (incorrect) The correct answer is a(a ). i x (y + ) = x y + (incorrect) The correct solution is x (y + ) = x y. iv) ( x)( y) = xy (incorrect) The correct solution is: ( x)( y) = xy 9 v) (x + l)(y + ) = x + y + (incorrect) The correct solution is: (x + )(y + ) = (x + )y + (x + ) = xy + y + x +.4 PROBLEMS. Find the values of the following expressions at the given variable values. i) x + 7x at x = (x + y) xy + y at x =, y = 
16 Math Review Kit: Section i (x % y) at x =, y = iv) & x % y & at x =, y = v) a bc at a =, b = 8, c = ab. Simplify the following: i) ( 6) i + 8[6 9(8 0)] iv) 0 {7 + [ (9 + )]} v) {( + 9) [ ( + 6 ) + 6( 7)]}. Simplify the following: i) x + x (4x 6x) a (a + b) + (a b) i x + y z (x y + z) (x + y) + (x z) iv) x{x y[z + (x y)]} v) {x + y [x (x + y)]} x y vi) xy + [x y + (x + y)(x y)] 6
17 SECTION 4 ALGEBRA OF FRACTIONS In this section, the algebraic operations discussed in section will be applied to fractions. Percentages and rounding procedure will be discussed as well. 4. THE LOWEST FORM OF FRACTIONS Definition (Fraction). An expression of the form a/b, b =/ 0, is called a fraction. Division by zero is not allowed. Since a/b = c means a = bc, a fraction like a/0 is not defined as we cannot find a unique number c such that a = 0 c. In the notation of infinity, we may say that if a > 0 then a/0 = and a/0 = . The fractions 0/0, /,  /,  /, and /, are undefined. They are known as indeterminate forms. A few other indeterminate forms are 0 and. It will be convenient to place fractions over a common denominator for addition or subtraction purposes. Definition (Equality of Fractions). We define a/b = c/d if ad = bc. 4 For example = because 6 = 4. 6 Definition gives rise to the following rule of fractions. Rule : Given a fraction a/b, the numerator (a) and denominator (b) can be multiplied or divided by a nonzero number without changing its value, i.e., a/b = ka/kb, k =/ 0. Example : i) = = = = Definition (Lowest Form of a Fraction). A fraction, a/b, is said to be of lowest form if a and b have no common factor. For example, /, /7, /9, / are fractions in lowest form, whereas, 6/4, 0/0, ab/bd, ab /a c are not. (A Factor is a number or variable that can be divided into both the numerator and the denominator such as to reduce the value of each.) Example : Reduce the fractions to their lowest form: 0 6a b i),, i, iv) 6 9ab 7a c 8ab c 4
18 Math Review Kit: Section 4 i) = 6 [Writing the numerator and denominator as a product of prime factors] = 4 [Dividing numerator and denominator by common factors ] 0 = = 6a b a a b i = = 9ab a b b a b 7a bc 7 a a b c iv) = = 8ab c 7 a b b c c a 4bc 4. CONVERTING FRACTIONS TO A COMMON DENOMINATOR Rule : The fractions a/b and c/d can be converted to an equivalent pair of fractions: ad/bd and bc/bd, with a common denominator bd. Sometimes it is convenient to convert into fractions with lowest common denominator. Example : Place the following fractions over a common denominator. i),, i, 6 i) = 6 is a common denominator. The fractions are then: 6 6 ', (i.e.,, 6 ' ) 6 6 = ' The fractions are, (, 8 6 ' ) 8 4
19 Algebra of Fractions i = ' 66 The fractions are,, ( ), ( 4 ' ) 4 Example 4: Place the fractions in example over the lowest common denominator. i), 6 6, 6 have 6 as a common denominator which is lowest. 4 The fractions are,. 6 6 i, have as the lowest common denominator. 6 The fractions are,. Definition 4 (Proper and Improper Fractions). A fraction in which the denominator is greater than the numerator is called a proper fraction. If the denominator is less than the numerator, it is called an improper fraction. An improper fraction can be changed into a mixed number which is the sum of an integer and a proper fraction. Example : Convert the following into mixed numbers. i) 4 0 So that = 4 + or 4 4
20 Math Review Kit: Section So that = 4. ALGEBRA OF FRACTIONS Rule : i) Sum (Difference) of Fractions. a b c +_ = d ad ± bc bd To find the sum (difference) of two fractions, place them over the lowest common denominator and add (subtract) the numerator. Example : i) + 4 Solve the following: = 8 0 % 0 ' 8 % 0 ' 0 ' 0 & 7 8 ' 6 & 4 6 [Note that 6 is the lowest common denominator] ' & 4 6 ' &9 6 i a b % ac 4bd 44
21 Algebra of Fractions Though Rule (i) can be used, we may also see that bd is the lowest common denominator. a b 4d 4d % ac 4bd a 4d bd % ac bd ' 8ad % ac bd ' a(8d % c) bd Rule 4: Multiplication of Fractions a b c d ' ac bd Example 7: Solve the following: i) 9 ' ' 0 ' 76 a b b a ' a b ' a b b b a b a a ' b a (Note here the use of Section 4. [definition ] in reducing to a lowestform fraction.) Rule : Division of Fractions a b c d ' a b d c ' ad bc 4
22 Math Review Kit: Section 4 Example 8: Solve the following. i) 8 ' 8 ' 6 ' a b c a b ' a b a b c ' a b 6bc ' a c 4.4 PERCENTAGES Definition (Percentage). Percent, denoted by %, is a fraction with 00 as the denominator. Example 9: Convert the following percentages into fractions and decimals. i) 8% 8 8% = = 00 8 Also, = /% % ' 00 ' 00 ' 00 Also, 00'.0 00 '.00 Example 0: Convert the following fractions to percentages: i) 46
23 Algebra of Fractions ' 0 0 ' ' 60% 6 In this case, it is difficult to convert the denominator to 00. Therefore convert the fraction to a decimal then multiply and divide by ' ' ' 8 % _ (Note that above the last figure of decimal indicates that figure repeats to infinity; thus.8 is really.8... ). 4. ROUNDING OFF A DECIMAL Rule of Rounding. To round off the decimal at any position, add one if the next decimal digit to the right is or more, or add nothing if the next decimal digit to the right is less than. For example: Rounding off.769 to two decimal places gives.77. Rounding off.874 to three decimal places gives.87. Rounding off to two decimal places gives 8.8. Rounding off to nearest cents Example : Round off to nearest cents. i) of $ Now 6 '.8 Multiplication by 00 will shift the decimal two places to the right. Therefore, round off at the fourth decimal place. Hence 00 =.8 00 = $
24 Math Review Kit: Section 4 of $ Now, = Multiplying by 000 will shift the decimal three places to the right. Therefore, round off at the fifth place ' ' $4.86 (Note: In Mortgage Finance calculations, payments are always rounded up, even if the decimal digit is less than.) 4.6 COMMON MISTAKES IN THE USE OF FRACTIONS x % y i) ' x % y (wrong) The correction solution is: x % y ' x % y a b c d ' a c bd (wrong) In fact, a b c d ' a b c d ' a b d c ' ad bc i 7 8 ' 7 7 ' 0 4 ' 0 (wrong) 7 8 ' 7 7 ' 4 Since 7 7 ' 48
25 Algebra of Fractions 4.7 PROBLEMS. Reduce the following to lowest form: 4 6 i) i iv) abc bcd. Place the fractions over their lowest common denominator: 7,, 7, 4 i) i 6 9, 6 a b, c iv) v) bd x yx, y xz. Insert one of <, = or > in the following: i) i & & iv) & 9 & 0 8 Hint: Note: Convert to common denominator and compare numerators. > means greater than ; < means less than. 4. Solve the following and reduce to lowest form: % 7 7 % i) i 4 8 & 6 iv) 4 % 6 & a b % c x % y v) vi) & x & y v bd z z a vi ix) bc x y b x) d 9 /6 a x xi) x xi a xiv) 8 /7 b bx xy zu x u. Convert the following percentages to fractions: i) % % i % iv) %
26 Math Review Kit: Section 4 6. Convert the following fractions to percentages: i) i iv) 8 7. Round off the following to the nearest cent: i) i of $000 over $0 8 of a million dollars iv) of $0, Simplify the following: [ 8 9 & ( & )] i) { 4 % [( & ) & ( & 6 )]} i { x % [x % y & (x & y)]} iv) ( a & b )( 4 & a 6 ) 40
27 SECTION ALGEBRA OF EXPONENTS In section, we introduced terms like x and x, referred to as exponential expressions. In this section we study the algebra of exponential expressions and numbers. The compound interest formula is discussed as well.. MEANING OF EXPONENT n An expression, symbolized by a, is called an exponential number or expression. The nonzero number a is called the base, and the number n is called the power or exponent. The interpretation of n depends on whether n is a positive integer, zero, negative integer, fractional or irrational number. Definition : n If n is a positive integer, then a means a multiplied by itself n times. That is, n a = a a a... a [n times] For example, = = 8. Similarly ( ) = ; ( ) = /8; ( ) = ; and 8 6 =. n Definition : If the exponent is a negative integer, then we define a = a n For example, = = 8 ( )  Similarly = ; = = = Definition : a =. Definition is a consequence of definitions and, as we shall see later in this section. /n n Definition 4: If n is a positive integer, then we define a as a number b such that b = a. In other n a /n th words, a is n root of a, and is denoted by. 4 Thus, a, which is also written as a, is called a square root of a, a is called a cube root of a and a is n a th th called a fourth root of a and so on. Note that we called as a n root and not the n root because there th may be more than one n root. For example 4 has two roots, = 4 and = 4. 
28 Math Review Kit: Section Example : Simplify the following: i) 8 8 i iv) / /4 / / / i) 8 = because = 8 /4 4 8 = because = 8 / i = = because = / / iv) = = because =. / As in parts i and iv), the same rule applies to negative fractional exponents as to negative integral exponents. (Refer to definition.) /n Care has to be taken when, in a, a is negative. Then the roots are not defined when n is even. However, / when n is odd, a root may exist. For example, () or & is not defined in the context of real numbers as there is no number b such that b =. But & =  as () = . For the sake of simplifying /n matters, we shall assume that a > 0 whenever finding a root of a is involved. Also, a would mean the th th positive n root of a. Talking of n roots, the following theorem, which generates many irrational numbers, will not be out of place. Theorem : If a is a prime number, then for every n where n =,,, 4,..., is an irrational number. n a Thus,,, 7, 4, are all irrational numbers. Example : Use your calculator, if possible, to check the following: i) ' ' i 4 ' iv) &9 ' & &.07 p/q th Definition : Let p/q be a fraction, then a is defined as the q root of a raised to the power p. q p/q a p /q p Symbolically, a = ( ) or (a ) 
29 Algebra of Exponents Example : / i) 4 = ( 4) = = 8 i 4/ = ( ) = = 8 / = = = = / ( ) iv) / (0.09) = ( 0.09) = (0.) =.07 n The exponential expression, a, with an irrational exponent n is difficult to compute because of infinite decimal expansion of irrational numbers. However, an approximation of the irrational exponent by a finite decimal, the number of digits depending upon the degree of approximation desired, can give a near value of a n. Example 4: i) = = 9.7 [Using a calculator and rounding off at third decimal] B = = 8.87 [Using a calculator and rounding off at third decimal]. ALGEBRA OF EXPONENTS n Now that a is defined for any real number n and a =/ 0, we are set to go into the algebra of exponents. Recall that whenever n is a fraction, we assume that a > 0. Let m and n be arbitrary numbers and a and b be nonzero numbers. m n m+n Rule : aa = a m n mn Rule : (a) = a Rule : (ab) = ab n n n a n Rule 4: ( ) = b a n b n a m Rule : = a m n a n 
30 Math Review Kit: Section Example : Simplify the following: i) ' 4 4 ' 8 [using rule ] 8 = [using rule ] 4 = = ' 4 ' 7 7 [using rule ] ' = 0 This example gives us an opportunity to check that a = for a =. 7 7 ' 7&7 ' 0 [using rule ] 0 Hence = = i (a b ) (a ) (b) (ab) 4 ' a a bb 4 a 4 b 4 [using rule ] ' ' a b 6a 4 b 4 6 ab & [using rule ] [using rule ] 4
31 Algebra of Exponents ' a 6b 4 iv) (x yz ) (x yz ) ' (x yz ) x yz 4 ' (x ) y (z ) x yz 4 [using rule ] ' 4x 4 y z 6 x yz 4 [using rule ] = 4xyz [using rule ] v) ab c x y ' (ab c ) (x y ) [using rule 4] ' a b 4 c 6 x 6 y 4 [using rules and ]. COMPOUND INTEREST The compound interest formula is an application of exponential expressions. Example 6: A person borrows $000 at 8% interest, compounded annually. What is the amount due after years? Figure 
32 Math Review Kit: Section The interest rate is $0.08 per dollar per year. Interest during first year = 000(.08) Amount due after one year = Principal + interest = (.08) = 000( +.08) Now, this amount becomes the principal at the beginning of the second year, then, interest during second year = 000( +.08)(.08). Amount due after two years = 000( +.08) + 000( +.08)(.08) = 000( +.08)[ +.08] = 000( +.08) This amount is the principal for the third year. Then, interest during the third year = 000( +.08) (.08). Amount due after years = 000( +.08) + 000( +.08) (.08) = 000( +.08) [ +.08] = 000( +.08) = 000(.08) = (using a calculator) = $9.7 In general, if an amount P, called principal, is invested at the interest rate of i per dollar per year compounded yearly, then the amount A due after n years is given by: A = P( +i) n This is the compound interest formula. Example 7: Solve example 6 using the compound interest formula. Here P = 000, i =.08, n = Therefore, A = 000( +.08) = = $9.7.4 PROBLEMS. Compute the following (do not give decimal answers): i) 4 7 i iv) ( v) (0.06) /4 49 )&/ / 4/ 6
33 Algebra of Exponents. Evaluate the following: 7 9 i) ( ) ( ) i / &/ / / (&8) / (&7) 4/ & iv) v) 4 / (&) / &/ &/. Approximate the irrational exponent by a threedigit decimal and compute using a calculator: i) B i e 4. Simplify: (x ) (x ) x yz i) ( ) i (x) (x ) axb (a ) (b ) (ab) iv) x[x + (x x)] v) (x y) vi) (x y) (xyz) / / / /. If $,000 is invested for years at 8% compounded yearly, what will this investment be worth at the end of this time? 6. On January, 97, a man incurs a debt of $,000 at % compounded annually. If the lender demands payment on January, 979, how much must the borrower pay? 7. As a part of his retirement savings program, a person sets aside $8,000 in a deferred savings account paying 9% compounded annually. What is the maturity value of this account after 0 years when he becomes 6? 7
34 SECTION 6 LINEAR FUNCTIONS AND INTERPOLATION 6. LINEAR INTERPOLATION Suppose two points (x,y ) and (x,y ) lie on a straight line and we want to find the value of y for a given x between x and x such that the point (x,y ) lies on the straight line. This process is called linear interpolation., If x is outside x and x, then it is called intrapolation. In both the cases, the two point formula as indicated in the following example is used. Example : An investment of $000 for years at 6% interest compounded annually yields $8. and at 8% compounded annually yields $469.. Use linear interpolation to find the amount at 7% compounded annually. Consider the points (6, 8.) and (8, 469.). Substituting these points for (x,y ) and (x,y ) into the two point formula and letting x = 7, we get: y & y y y = (x & x x & x ) [The two point formula] 469. & 8. y 8. = (7 6) 8 & 6.0 = () = 6. Hence y = = Therefore, at 7%, the interpolated amount is $ PROBLEMS. On a piece of land, the application of gallon of fertilizer yields bushels of a certain crop and the application of gallons of fertilizer yields bushels of the crop. Estimate the yield if gallons of fertilizer were used, using linear (twopoint) interpolation. 6
35 SECTION 7 SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES In section 6, we discussed linear equations in two variables. In this section, we will discuss the algebraic aspects of linear equations. 7. LINEAR EQUATIONS IN ONE VARIABLE A linear equation in one variable, x, is of the form: ax + b = 0, a =/ 0. (A) Definition (Solution). A value of x is said to be the solution of equation (A) if it satisfies the equation. Example : i) Solve x = 0 and check to see that the solution satisfies the equation. Using the laws of equality, we have: x + = (adding on both sides) x = This is also known as transposing on the right hand side of the equation by changing its sign. Simplifying further, we get: x = (dividing both sides by ) Substituting x = in the original equation, we get: ( ) = 0 = 0 0 = 0 Hence, x = is the solution of the given equation. 7
36 Math Review Kit: Section 7 Solve x + = x + and check the solution. Adding (x ) on both sides we get: x + x = x + x x = 0 x = (dividing the equation by ) Substituting x = in the original equation, we get: () + = + + = 0 0 = 0 Hence, x = is the solution of the given equation. 7. PROBLEMS. Solve for x: i) (x +) = x + x + = x + 0 i (x + ) +( x) = 7
37 SECTION 8 IDENTITIES AND FACTORIZATION In this section, we introduce identities and use them in the factorization of algebraic expressions. Factorization of quadratic expressions is discussed in detail. 8. THE IDENTITIES Definition (Identity). An identity is an equation involving two algebraic expressions which is satisfied by all possible values of the variables in the equation. For example, x = x + x; x + x = x(x + ). We list below a few common identities which can be confirmed by using the distributive law upon the left hand side (L.H.S.): (x + y) = x + xy + y () (x y) = x xy + y () (x + y)(x y) = x y () (x + y) = x + x y + xy + y (4) (x y) = x x y + xy y () (x + y)(x xy + y ) = x + y (6) (x  y)(x + xy + y ) = x y (7) As an illustration, we check identity () L.H.S. = (x + y) = (x + y)(x + y) = x(x + y) + y(x + y) [using the distributive law] = x + xy + yx + y = x + xy + y = R.H.S. Example : Simplify the following: i) (x + ), (a ), i (m + )(m ), iv) (x + ) 8
38 Math Review Kit: Section 8 i) Using identity (), we have: (x + ) = (x) + [(x) ()] + = 4x + x + 9 Using identity (), we get: (a ) = (a) [(a) ()] + = 9a 0a + i In this case, identity () applies. Thus: (m + )(m ) = m = m iv) Using identity (4), we have: (x + ) = x + (x ) + (9x) + = x + 9x + 7x FACTORIZATION Definition (Factorization). By factorization of an algebraic expression, we mean writing the expression as a product of two or more terms, called its factors. Obviously, the factors and the algebraic expression itself are not used in factorization. For example: x + 0 = (x + ) and x y = (x + y)(x y) Note that these two examples are also identities. The identities and the distributive law are extensively used in factorization. Example : Factorize the following expressions: i) xy + xz + x i a b + ab + ab x y + 6x y + 7x y + x y 8
39 Identities and Factorization iv) 7m 8n i) We note that x is a common factor in each term. Invoking the distributive law, we factor out x: xy + xz + x = x(y + z + ) Factoring out ab and using the laws of exponents: a b + ab + ab = ab(a + + b) i Factoring out x y and using the laws of exponents: x y + 6x y + 7x y + x y = x y(x + y + 9y + ) iv) 7m 8n = (m) (n) = (m n)[(m) + (m n) + (n) ] [using the identity (7)] = (m n)(9m + 6mn + 4n ) 8. PROBLEMS. Factorize the following: i) 8m n 6 a + 7 i iv) v) 8a b abc + a b + 0abc x y % 4 x y 8
40 Solutions
41
42 SOLUTIONS Solutions to the Problems in Section 8. The rational numbers are: 6, ,,, , i) + ( 6) = 7 i iv) 6 + ( 9) = + ( 8) = 7 ( ) = v) 7 (+6) = vi) v (8)( ) = 04 + (0) = vi ( )( 7) = ix) (&6) & (&7) (&)(&) ' & & (&) ' 9 ' Solutions to the Problems in Section. The additive inverse (a.i.) of 8 is 8 because 8 + ( 8) = 0 and = 0. Similarly the a.i. of 9 is 9. The a.i. of 0 is 0. The a.i. of 6 is 6. The a.i. of is. The a.i. of 9 is 9.. The multiplicative inverse (m.i.) of is because = and =. Similarly the m.i. of is, the m.i. of 9 is, the m.i. of 9 is and the m.i. of is. 9 9 Solutions
43 Math Review Kit: Solutions. i) True True i False because a(b + ) = ab + a iv) True v) True vi) True v False because (a + b)( c) = a( c) + b( c) = a ac + b bc Solutions to the Problems in Section. i) ( ) + 7( ) = ( )( ) + ( 4) = 4 4 = 8 6 = 8 ( + ) () + = = = = i iv) [ % (&)] ' (9&) ' 8 ' 64 ' & % & '& % 9 & '&8 v) 8 (&) 8 ' 4 8(&) 6 ' 4 &8 6 ' &96 4 '&4. i) = 9 + = = = ( 6) = 4 ( ) = + = 4 i + 8[6 9(8 0)] = + 8[6 9( )] = + 8[6 ( 8)] = + 8[6 + 8] = + 8[4] = + 9 = 94 Solutions
44 Math Review Kit: Solutions iv) 0 {7 + [ (9 + )]} = 0 {7 + [ 6]} = 0 {7 8} = 0 { } = 0 + = v) {( + 9 ( [ ( + 6 ) + 6( 7)]} = {( + 9) [ (9 ) + 6( 4)]} = {8 [ ( ) 4]} = {8 [ 9]} = {8 + 8} = {46} = 9. i) x + x (4x 6x) = 4x 4x + 6x = 6x a (a + b) + (a b) = a a b + a b = b i x + y z (x y + z) (x + y) + (x z) = x + y z x + y z x y + x z = x x x + x + y + y y z z z = x + y z iv) x{x y[z + (x y)]} = x{x y[z + x y]} = x{x yz yx + y } = x xyz x y + 6xy v) {x + y [x (x + y]} x y = {x + y [x x y]} x y = {x + y [x y]} x y = {x + y x + y} x y = 4y x y = y x vi) xy + [x y + (x + y)(x y)] = xy + [x y + x(x y) + y(x y)] = xy + [x y + x xy + xy y ] = xy + x y + x xy + xy y = x y + xy xy + xy + x y = x y xy + x y Solutions
45 Math Review Kit: Solutions Solutions to the Problems in Section 4. i) i 4 ' ' 49 ' ' ' ' 9 6 iv) abc bcd ' a b c c b c d ' ac d. i) Lowest common demoninator (l.c.d.) is 7 = 9, hence the fractions are: 7 and 7 7 or 9 and 4 9 The l.c.d. is 7. The fractions are:, 7, 4 0 or 7, 7, 7 i The l.c.d. is 9. The fractions are: 6 9, 6 or 80 9, 6 9 iv) The l.c.d. is bd. The fractions are: a b b d, x x yz x, c bd or ad bd, c bd v) The l.c.d. is xyz. The fractions are: y y xz y or x xyz, y xyz. i) < 8 as ' > 7 as 0 ' 70 and 7 ' 0 70 Solutions4
46 Math Review Kit: Solutions i iv) & > & 7 as & '&4 and & 7 '& & 9 '&0 8 as & 9 '& 9 '& i) % 7 ' 7 % ' 7 % 7 % 4 ' 4 4 % 4 ' 7 4 ' ' i 8 & 6 ' 88 & ' & ' 7 88 iv) v) 4 % 6 & ' 0 68 % 9 68 & 8 68 ' 0 % 9 & 8 68 a b % c bd ' ad bd % c bd ' ad % c bd ' 68 ' 8 vi) x % y z & x & y z ' x % y & (x & y) z ' x % y & x % y z ' y z v vi ix) x) xi) x 7 8 ' 4 40 ' ' 0 ' a bc x y b d ' axb bcyd ' ax cyd 9 % ' 9 ' 6 9 ' /6 8 ' 6 8' 6 8 ' 48 /7 ' 7 ' 7 ' ' xi a x b a bx ' a x b bx a ' a x b ab ' ax ' ax Solutions
47 Math Review Kit: Solutions xiv) xy zu x u ' xy zu u x ' xyu zux ' yu zx. i) i iv) % ' 00 ' 0 4 % ' /4 00 ' 4 00 ' % ' /4 00 ' 4 00 ' 400 % ' 00 ' ' 4 ' 4 6. i) i iv) = 00% = 4% = 00% = 40% = 00% = 66.6% = 66 % 8 8 = 00% =.% = % 7. i) i iv) of $000 =. 000 = $.00 8 of $0 = = $8.4 of $,000,000 =.,000,000 = $,. 9 of $0,000 = ,000 = $, i) [ 8 9 & ( & )] = [ 8 9 & 6 ]as & ' 6 & 6 ' 6 Solutions6
48 Math Review Kit: Solutions = = = [ 6 8 & 8 ] [ 8 ] ' { 4 % [( & ) & ( & 6 )]} = { 4 % [ & 6 ]} as ' and ' 6 = { 4 & 7 6 }as ' 4 6 { & 4 } ' = (& ) = & 4 i = = = = = = { x % [x % y & (x & y)]} { x % [x % y & x % y]} { x % [ x % 4 y]} { x % x % y} { 6 x % y} x % 4 y iv) ( a & b )( 4 & a 6 ) = = = a ( 4 & a 6 ) & b ( 4 & a 6 ) a 8 & a & b % ab 8 8 a & a & b % 8 ab Solutions to the Problems in Section. i) 4 ' (4) ' ( ) ' ' ' 9 Solutions7
49 Math Review Kit: Solutions 6 7 ' ( ) ' ' ' 9 i & 4 ' ' 4/ ( ) ' 4/ ' 4 6 iv) 49 &/ ' 49 / ' 49/ ' (7 ) / / ( ) 76/ 7 ' ' / 6/ ' 4 v) (0.06) ' [(.) 4 ] '. '.. i) ' 4 ' 6 ' & ' 9 4 ' ( ) ( ) ' 4 4 ' & ' i iv) / &/ / / ' ' & ' (&8) / (&7) 4/ ' (& ) / (& ) 4/ ' &6/ & / 4 / (&) / ( ) / (& ) / 6/ & 6/ ' & & 4 ' 4 8 & 8 ' 4 00 ' 8 0 ' 0 v) & &/ &/ ' / / / ' / / / ' ( % 8 % % ) ' / ' ' * Alternatively: ( ) / (& ) 4/ ( ) / & (& ) ' (&) (&) 4 / ( )(& ) i) = = = 4.78 (rounded value) B ' '.4 ' Solutions8
50 Math Review Kit: Solutions i e ' '.78 ' (i) (x )(x ) (x)(x ) ' x 7 x 6 ' x ' x ( x y axb ' x 6 y z a x b ' x y z ' xyz a b ab (i (iv) (v) (a ) (b ) ' a 6 b 6 ' a b (ab) a b x[x % (x & x)] ' x[x % x & x] ' x % x & x (x &/ y) ' [x &/ y / ] ' [x &/ ] [y / ] ' x & y / ' y / x (vi) (x y) / (xyz) &/ ' [(x ) / y / ][x &/ y &/ z &/ ] ' [x 4/ y / ][x &/ y &/ z &/ ] ' xy / z &/ ' xy / z /. P = 00, i =.08, n = A = 000( +.08) = = $ P = 000, i =., n = 4 A = 000( +.) 4 = = $ P = 8000, i =.09, n = 0 = 8000( +.09) 0 = = $8,98.9 Solutions to the Problems in Section 6. Let x denote the amount of fertilizer and y, the crop yield. Then the two points are (,) and (,). The question of the line passing through these points is: Solutions9
51 Math Review Kit: Solutions y & y ' y & y x & x (x & x ) [The Two Point Formula] y & ' & & (x & ) ' (x & ) y ' x & % [adding to both sides] y ' x % for x =, y ' () % ' 6 % ' 7 ' Hence, the estimated yield corresponding to gallons of fertilizer is / bushels. Solutions to the Problems in Section 7. i) (x + ) = x + = x + = x = 0 [adding ( ) to both sides] x = x + x + = x + 0 x + 7 = x + 0 Adding 7 and x both sides, x x = x x x = x = i (x + ) + ( x) = x x = x + = [subtracting from both sides] x = 0 x = 0 Solutions to the Problems in Section 8. i) 8m n = (9m) (n) = (9m + n)(9m n) [using identity ()] Solutions0
52 Math Review Kit: Solutions 6 a + 7 = (a ) + Using identify (6) we get: (a + )[(a ) a + ] 4 = (a + )(a a + 9) i iv) 8a b = (a) (b) = (a b)[(a) + a b + (b) ] [using identity (7)] = (a b)(4a + 0ab + b ) abc + a b + 0abc Factoring out ab, we get: ab(c + a + c ) v) xy + xy 4 Factoring out 4 x y, we get: 4 x y (x + y) Solutions
Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationBEGINNING ALGEBRA ACKNOWLEDMENTS
BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationP.E.R.T. Math Study Guide
A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT  A Math Study Guide 1. Linear Equations
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationTransition To College Mathematics
Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering
More informationMultiplication and Division Properties of Radicals. b 1. 2. a Division property of radicals. 1 n ab 1ab2 1 n a 1 n b 1 n 1 n a 1 n b
488 Chapter 7 Radicals and Complex Numbers Objectives 1. Multiplication and Division Properties of Radicals 2. Simplifying Radicals by Using the Multiplication Property of Radicals 3. Simplifying Radicals
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationSIMPLIFYING SQUARE ROOTS
40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationRules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER
Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More information126.7 87. 88. 89. 90. 13.2. Exponents, Roots, and Order of Operations. OBJECTIVE 1 Use exponents. In algebra, w e use exponents as a w ay of writing
SECTION 1. Exponents, Roots, and Order of Operations 227.72126.7100 50 87. 88. 89. 90. 1.2 6.20.010.05 Solve each problem. 91. The highest temperature ever recorded in Juneau, Alaska, was 90 F. The
More informationReview of Basic Algebraic Concepts
Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable.
More informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More informationACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011
ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationTo Evaluate an Algebraic Expression
1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationMATH0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationLESSON 6.2 POLYNOMIAL OPERATIONS I
LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order
More informationChapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationGCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.
GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationOPERATIONS WITH ALGEBRAIC EXPRESSIONS
CHAPTER OPERATIONS WITH ALGEBRAIC EXPRESSIONS Marvin is planning two rectangular gardens that will have the same width. He wants one to be 5 feet longer than it is wide and the other to be 8 feet longer
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationFACTORISATION YEARS. A guide for teachers  Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers  Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationCHAPTER 5 Roundoff errors
CHAPTER 5 Roundoff errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (125) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More informationWelcome to Basic Math Skills!
Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationFINDING THE LEAST COMMON DENOMINATOR
0 (7 18) Chapter 7 Rational Expressions GETTING MORE INVOLVED 7. Discussion. Evaluate each expression. a) Onehalf of 1 b) Onethird of c) Onehalf of x d) Onehalf of x 7. Exploration. Let R 6 x x 0 x
More informationHFCC Math Lab Arithmetic  4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic  Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationHow to get the Simplified Expanded Form of a polynomial, II
How to get the Simplified Expanded Form of a polynomial, II Nikos Apostolakis September 29, 2010 Recall. Recall that the distributive law states that multiplication distributes over addition and subtraction:
More informationUsing the Properties in Computation. a) 347 35 65 b) 3 435 c) 6 28 4 28
(1) Chapter 1 Real Numbers and Their Properties In this section 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS The properties of the real numbers can be helpful when we are doing computations. In this
More informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
More information2 : two cube. 5 : five cube. 10 : ten cube.
Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VI Instructor: Line #: 52920 Yasuyuki Kachi 6 Cubes February 2 Mon, 2015 We can similarly define the notion of cubes/cubing Like we did last time, 3 2
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More information