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1 1 Introduction Mathematics is an exciting field of study, concerned with structure, patterns and ideas. To fully appreciate and understand these core aspects of mathematics, you need to e confident and skilled in the rules and language of algera. Although you have encountered some, perhaps most or all, of the material in this chapter in a previous mathematics course, the aim of this chapter is to ensure that you are familiar with fundamental terminology, notation and algeraic techniques. 1.1 The real numers The most fundamental uilding locks in mathematics are numers and the operations that can e performed on them. Algera, like arithmetic, involves performing operations such as addition, sutraction, multiplication and division on numers. In arithmetic, we are performing operations on known, specific, numers (e.g. 5 8). However, in algera we often deal with operations on unknown numers represented y a variale - usually symolized y a letter ( e.g. _ a c a c c ). The use of a variale gives us the power to write general statements indicating relationships etween numers. But what types of numers can variales represent? All of the mathematics in this course involves the real numers and susets of the real numers. A real numer is any numer that can e represented y a point on the real numer line (Figure 1.1). Each point on the real numer line corresponds to one and only one real numer, and each real numer corresponds to one and only one point on the real numer line. This kind of relationship is called a one-to-one correspondence. The numer associated with a point on the real numer line is called the coordinate of the point. The word algera comes from the 9th-century Araic ook Hisâ al-jar w al-muqaala, written y al-khowarizmi. The title refers to transposing and comining terms, two processes used in solving equations. In Latin translations, the title was shortened to Aljar, from which we get the word algera. The author s name made its way into the English language in the form of the word algorithm π Figure 1.1 The real numer line Susets of the real numers The set of real numers contains some important susets with which you should e familiar. When we first learn to count, we use the numers 1,,,. These numers form the set of counting numers or positive integers. 1

2 1 Downloaded from Hint: Do not e confused if you see other textooks indicate that the set (usually referred to as the natural numers) does not include zero and is defined as {1,,, }. There is disagreement among mathematicians whether zero should e considered a natural numer i.e. reflecting how we naturally count. We normally do not start counting at zero. However, zero does represent a counting concept in that it is the asence of any ojects in a set. Therefore, some mathematicians (and the IB mathematics curriculum) define the set as the positive integers and zero. Figure 1. A Venn diagram representing the relationships etween the different susets of the real numers. The rational numers comined with the irrational numers make up the entire set of real numers. Adding zero to the positive integers (0, 1,,, ) forms the set referred to as the set in IB notation. The set of integers consists of the counting numers with their corresponding negative values and zero (,, 1, 0, 1,,, ) and is denoted y (from the German word Zahl for numer). We construct the rational numers y taking ratios of integers. Thus, a real numer is rational if it can e written as the ratio p q of any two integers where q 0. The decimal representation of a rational numer either repeats or terminates. For example, _ _ (the lock of six digits repeats) or _ (the decimal terminates at 5, or alternatively has a repeating zero after the 5). A real numer that cannot e written as the ratio of two integers, such as and, is called irrational. Irrational numers have infinite non-repeating decimal representations. For example, and There is no special symol for the set of irrational numers. real numers 10 irrational numers 1 5 π rational numers 1 integers 6 58,,, 1, 0, 1,,, 17 9 Tale 1.1 A summary of the susets of the real numers and their symols. Positive integers Positive integers and zero Integers Rational numers {1,,, } {0, 1,,, } {,, 1, 0, 1,,, } any numer that can e written as the ratio p q of any two integers where q 0 Sets and intervals If every element of a set C is also an element of a set D, then C is a suset of set D, and is written symolically as C D. If two sets are equal (i.e. they have identical elements), then they satisfy the definition of a suset and each would e a suset of the other. For example, if C {,, 6} and D {,, 6}, then C D and D C. What is more common is that a suset is a set that is contained in a larger set and does not contain at least one element of the larger set. Such a suset is called a proper suset and is denoted with the symol. For example, if D {,, 6} and E {, }, then E is a proper suset of D and is written E D. All of the susets of the real numers discussed earlier in this section are proper susets of the real numers, for example, and.

3 The symol indicates that a numer, or a numer assigned to a variale, elongs to (is an element of) a set. We can write 6, which is read 6 is an element of the set of integers. Some sets can e descried y listing their elements within rackets. For example, the set A that contains all of the integers etween and inclusive can e written as A {, 1, 0, 1, }. We can also use set-uilder notation to indicate that the elements of set A are the values that can e assigned to a particular variale. For example, the notation A {x, 1, 0, 1, } or A {x x } indicates that A is the set of all x such that x is an integer greater than or equal to and less than or equal to positive. Set-uilder notation is particularly useful representing sets for which it would e difficult or impossile to list all of the elements. For example, to indicate the set of positive integers n greater than 5, we could write {n n 5} or {n n 5, n }. The intersection of A and B (Figure 1.), denoted y A B and read A intersection B, is the set of all elements that are in oth set A and set B. The union of two sets A and B (Figure 1.), denoted y A B and read A union B, is the set of all elements that are in set A or in set B (or in oth). The set that contains no elements is called the empty set (or null set) and is denoted y. A B A B Figure 1. Intersection of sets A and B. A B Figure 1. Union of sets A and B. A B Some susets of the real numers are a portion, or an interval, of the real numer line and correspond geometrically to a line segment or a ray. They can e represented either y an inequality or y interval notation. For example, the set of all real numers x etween and 5, including and 5, can e expressed y the inequality x 5 or y the interval notation x [, 5]. This is an example of a closed interval (i.e. oth endpoints are included in the set) and corresponds to the line segment with endpoints of x and x An example of an open interval is x 1 or x ], 1[ where oth endpoints are not included in the set. This set corresponds to a line segment with open dots on the endpoints indicating they are excluded If an interval, such as x or x [, [, includes one endpoint ut not the other, it is referred to as a half-open interval. Hint: Unless indicated otherwise, if interval notation is used, we assume that it indicates a suset of the real numers. For example, the expression x [, ] is read x is any real numer etween and inclusive. Hint: The symols (positive infinity) and (negative infinity) do not represent real numers. They are simply symols used to indicate that an interval extends indefinitely in the positive or negative direction

4 1 Downloaded from The three examples of intervals on the real numer line given aove are all considered ounded intervals in that they are line segments with two endpoints (regardless whether included or excluded). The set of all real numers greater than is an open interval ecause the one endpoint is excluded and can e expressed y the inequality x, or x (, ). This is also an example of an unounded interval and corresponds to a portion of the real numer line that is a ray Tale 1. The nine possile types of intervals oth ounded and unounded. For all of the examples given, we assume that a. Interval notation Inequality Interval type Graph x [a, ] a x closed a x ]a, [ a x open a x [a, [ a x half-open a x ]a, ] a x half-open a x [a, [ x a half-open a x ]a, [ x a open a x ], ] x half-open x ], [ x open x ], [ real numer line Asolute value The asolute value of a numer a, denoted y a, is the distance from a to 0 on the real numer line. Since a distance must e positive or zero, then the asolute value of a numer is never negative. Note that if a is a negative numer then a will e positive. Definition of asolute value If a is a real numer, the asolute value of a is a if a 0 a a if a 0 Here are four useful properties of asolute value: Given that a and are real numers, then: 1 a 0 a a a a a _ a 0

5 Asolute value is used to define the distance etween two numers on the real numer line. Distance etween two points on the real numer line Given that a and are real numers, then the distance etween the points with coordinates a and on the real numer line is a which is equivalent to a Asolute value expressions can appear in inequalities, as shown in the tale elow. Inequality Equivalent form Graph x a x a x a x a a x a a x a x a or x a x a or x a a a a a a a a a Tale 1. Properties of asolute value inequalities. Properties of real numers There are four arithmetic operations with real numers: addition, multiplication, sutraction and division. Since sutraction can e written as addition (a a ( )), and division can e written as multiplication ( a a ( 1 ), 0 ), then the properties of the real numers are defined in terms of addition and multiplication only. In these definitions, a is the additive inverse (or opposite) of a, and 1 a is the multiplicative inverse (or reciprocal) of a. Tale 1. Properties of real numers. Property Rule Example Commutative Property of Addition: a a x y y x Commutative Property of Multiplication: a a ( x )x x ( x ) Associative Property of Addition: (a ) c a ( c) (1 x ) 5x 1 (x 5x) Associative Property of Multiplication: (a)c a(c) (x 5y) ( 1 y ) (x) ( 5y 1 y ) Distriutive Property: a( c) a ac x ( x ) x x x ( ) Additive Identity Property: a 0 a y 0 y Multiplicative Identity Property: 1 a a _ 1 _ _ _ 8 1 Additive Inverse Property: a ( a) 0 6y ( 6y ) 0 Multiplicative Inverse Property: a 1 a 1, a 0 ( y ) ( 1 _ y ) 1 Note: These properties can e applied in either direction as shown in the rules aove. 5

6 1 Downloaded from Exercise 1.1 In questions 1 6, plot the two real numers on the real numer line, and then find the distance etween their coordinates. 1 5; _ ; 11 1.; 6 7; 5_ 5 ; 6 5_ 6 ; 9_ In questions 7 1, write an inequality to represent the given interval and state whether the interval is closed, open or half-open. Also state whether the interval is ounded or unounded. 7 [ 5, ] 8 ] 10, ] 9 [1, [ 10 ], [ 11 [0, [ 1 [a, ] In questions 1 18, use interval notation to represent the suset of real numers that is indicated y the inequality. 1 x 6 1 x 8 15 x x 1 17 x 5 18 x In questions 19, use inequality and interval notation to represent the given suset of real numers. 19 x is at least 6. 0 x is greater than or equal to and less than x is negative. x is any positive numer less than 5. In questions 8, state the indicated set given that A {1,,,, 5, 6, 7, 8}, B {1,, 5, 7, 9} and C {,, 6}. A B A B 5 B C 6 A C 7 A B 8 A B C In questions 9, use the symol to write a correct statement involving the two sets. 9 and 0 and 1 and and In questions 6, express the inequality, or inequalities, using asolute value. 6 x 6 x or x 5 x 6 x 1 or x 1 In questions 7, evaluate each asolute value expression In questions 6, find all values of x that make the equation true. x 5 x 5 6 x 10 6 x 5 1 6

7 1. Roots Roots and radicals (surds) If a numer can e expressed as the product of two equal factors, that factor is called the square root of the numer. For example, 7 is the square root of 9 ecause Now 9 is also equal to 7 7, so 7 is also a square root of 9. Every positive real numer will have two real numer square roots one positive and one negative. However, there are many instances where we want only the positive square root. The symol 0 (sometimes called the root or radical symol) indicates only the positive square root often referred to as the principal square root. In _ words, the square roots of 16 are and ; ut, symolically, 16. The _ negative square root of 16 is written as 16, and when oth square roots _ are wanted we write 16. When a numer can e expressed as the product of three equal factors, then that factor is called the cue root of the numer. For example, is the cue root of 6 ecause ( )( )( ) 6. This is written symolically as 6. In general, if a numer a can e expressed as the product of n equal factors then that factor is called the nth root of a and is written as n a. n is called the index and if no index is written it is assumed to e a, therey indicating a square root. If n is an even numer (e.g. square root, fourth root, etc.) then the principal nth root is positive. For example, since ( )( )( )( ) 16, then is a fourth root of 16. However, the principal fourth root of 16, written _ 16, is equal to. Radicals (surds) Some roots are rational and some are irrational. Consider the two right triangles in Figure 1.5. By applying Pythagoras theorem, we find the length of the hypotenuse for triangle A to e exactly 5 (an integer and rational numer) and the _ hypotenuse for triangle B to e exactly 80 (an irrational numer). An irrational root e.g. 80,, 10, is called a radical or surd. The only way to express irrational roots exactly is in radical, or surd, form. It is not immediately ovious that the following expressions are all equivalent. 80, 0, 16 _ 5 16, _ 10, 10 _ 8 10, 5, Square roots occur frequently in several of the topics in this course, so it will e useful for us to simplify radicals and to recognise equivalent radicals. Two useful rules for manipulating expressions with radicals are given elow. A x x x 9 16 x _ 5 x _ 5 _ x 5 x 5 Figure y 8 y 16 6 y 80 y _ 80 _ y 80 _ Hint: The solution for the hypotenuse of triangle A in Figure 1.5 involves the equation x 5. Because x represents a length that must e positive, we want only the positive square root when taking the square root _ of oth sides of the equation i.e. 5. However, if there were no constraints on the value of x, we must rememer that a positive numer will have two square roots and we would write x _ 5 x 5. y B 7

8 1 Downloaded from Simplifying radicals For a 0, 0 and n, the following rules can e applied: 1 n a n n _ n a a _ n n a Note: each rule can e applied in either direction. Example 1 Simplify each of the radicals. _ a) 5 5 ) 18 c) 8 a) _ d) _ 6 6 Note: A special case of the rule n a n n a when n is a a a. ) c) _ 8 8 _ 16 d) _ 1 _ 18 _ _ The radical can e simplified ecause one of the factors of is, and the square root of is rational (i.e. is a perfect square). _ _ 6 _ 6 6 Rewriting as the product of and 8 (rather than and 6) would not _ help simplify ecause neither nor 8 are perfect squares. Example Express each in terms of the simplest possile radical. a) 18 ) 80 c) 5 a) ) d) 1000 Note: is a factor of 80 and is a perfect square, ut 16 is the largest factor that is a perfect square. c) 5 5 _ 5 d) We prefer not to have radicals in the denominator of a fraction. Recall from Example 1a), the special case of the rule n a n n _ a when n is a a a. The process of eliminating irrational numers from the denominator is called rationalising the denominator. 8

9 Example Rationalise the denominator of each expression. a) _ a) _ _ _ ) _ 7 _ 10 _ ) _ 10 Exercise 1. In questions 1 9, express each in terms of the simplest possile radical. _ _ _ In questions 10 1, completely simplify the expression. _ _ In questions 1 19, rationalise the denominator, simplifying if possile. 1 _ 1 15 _ _ _ 1. Exponents (indices) Repeated multiplication of identical numers can e written more efficiently y using exponential notation. Exponential notation If a is any real numer (a ) and n is a positive integer (n ), then a n a a a a n factors where n is the exponent, a is the ase and a n is called the nth power of a. Note: n is also called the power or index (plural: indices). Integer exponents We now state seven laws of integer exponents (or indices) that you will have learned in a previous mathematics course. Familiarity with these rules is essential for work throughout this course. Let a and e real numers (a, ) and let m and n e integers (m, n ). Assume that all denominators and ases are not equal to zero. All of the laws can e applied in either direction. 9

10 1 Downloaded from Tale 1.5 Laws of exponents (indices) for integer exponents. Hint: It is important to recognise the difference etween exponential expressions such as ( ) and. In the expression ( ), the parentheses make it clear that is the ase eing raised to the power of. However, in the negative sign is not considered to e a part of the ase with the expression eing the same as () so that is the ase eing raised to the power of. Hence, ( ) 9 and 9. Property Example Description 1. m n m n x x 5 x 7 multiplying like ases. _ m n m n w7 w w5 dividing like ases. ( m ) n mn ( x ) x ( ) x 9 x a power raised to a power. (a) n a n n (k) k 6k the power of a product 5. ( a ) n an n ( y ) y y 9 the power of a quotient 6. a 0 1 (t 5) 0 1 definition of a zero power 7. a n 1 a n definition of a negative exponent The last two laws of exponents listed aove the definition of a zero exponent and the definition of a negative exponent are often assumed without proper explanation. The definition of a n as repeated multiplication, i.e. n factors of a, is easily understood when n is a positive integer. So how do we formulate appropriate definitions for a n when n is negative or zero? These definitions will have to e compatile with the laws for positive integer exponents. If the law stating m n m n is to hold for a zero exponent, then n 0 n 0 n. Since the numer 1 is the identity element for multiplication (Multiplicative Identity Property) then n 1 n. Therefore, we must define 0 as the numer 1. If the law m n m n is to also hold for negative integer exponents, then n n n n 0 1. Since the product of n and n is 1, they must e reciprocals (Multiplicative Inverse Property). Therefore, we must define n as 1 n. Rational exponents We know that and 0 1 and 1 _ 1, ut what meaning are we to give to 1_? In order to carry out algeraic operations with expressions having exponents that are rational numers, it will e very helpful if they follow the laws estalished for integer exponents. From the law m n m n, it must follow that 1_ 1_ 1_ 1_ 1. Likewise, from the law ( m ) n mn, it follows that ( 1_ ) 1_ 1. Therefore, we need to define 1_ as the square root of, or more precisely as the principal (positive) square root of, that is. We are now ready to use radicals to define a rational exponent of the form 1 n where n is a positive integer. If the rule ( m ) n mn is to apply when m 1 n, it must follow that ( 1_ n ) n n_ n 1. This means that the nth power of 1_ n is and, from the discussion of nth roots in Section 1., we define 1_ n as the principal nth root of. Definition of 1 n If n, then n 1 is the principal nth root of. Using a radical, this means n 1 n 10

11 This definition allows us to evaluate exponential expressions such as the following: ; ( 7) 1_ 7 ; _ ( 81) 1 1_ Now we can apply the definition of n 1_ and the rule ( m ) n mn to develop a rule for expressions with exponents of the form not just n 1 ut of the more general form m n. _ m n m n 1_ ( m ) n 1_ n m ; or equivalently, m n n 1_ m ( n 1_ ) m ( n ) m This will allow us to evaluate exponential expressions such as 9 _, ( 8) 5_ and 6 5_ 6. Definition of rational exponents If m and n are positive integers with no common factors, then m n n _ m or ( n ) m If n is an even numer, then we must have 0. The numerator of a rational exponent indicates the power to which the ase of the exponential expression is raised, and the denominator indicates the root to e taken. With this definition for rational exponents, we can conclude that the laws of exponents stated for integer exponents in Section 1. also hold true for rational exponents. Example Evaluate and/or simplify each of the following exponential expressions. a) (xy ) ) (xy ) c) ( ) d) (a ) 0 e) ( ) 1_ 9 _ f) a _ a 5 5 g) ( ) _ 5 h) 8 _ i) ( 1_ x y ) (x y ) 1 j) _ a a k) a) (xy ) x (y ) 8x y 6 ) (xy ) x (y ) x y 6 c) ( ) _ 1 ( ) 1 8 d) (a ) 0 1 (x y) (x y) e) ( ) 1_ 9 _ _ ( ) 1_ _ _ 6_ 7 f) _ a a 5 5 _ a ( 5) 5 a g) ( ) _ 5 [( 5 ] _ 5 ( ) _ 1 ( ) _ 1 16 h) 8 _ 8 _ 6 or 8 _ ( 8 ) () or 8 _ ( ) _ i) ( 1 x y ) (x y ) 1 ( x 6 y 8 )(x y ) x 6 y _ 8 x y

12 1 Downloaded from j) _ 1_ a a _ (a ) (a ) 1 1 _ 1 1 (a ) 1 1_ (a ) 1_ a Note: Avoid an error here. n _ a n a n _. Also, a a and a a. (x y) k) (x y) (x y) ( ) (x y) Note: Avoid an error here. (x y) n x y. Although (x y) x x y 6x y xy y, expanding is not generally simplifying. Exercise 1. In questions 1 6, simplify (without your GDC) each expression to a single integer _ 9 _ 6 _ 8 _ 5 _ 5 6 ( ) 6 In questions 7 9, simplify each expression (without your GDC) to a quotient of two integers. 7 _ ( 8 ) _ 7 ) _ 8 ( 9 16 ) 1_ 9 ( 5 In questions 10 1, evaluate (without your GDC) each expression. 10 ( ) 11 (1) ( ) In questions 1 0, simplify each exponential expression (leave only positive exponents). 1 ( a ) 15 ( a ) 16 ( a ) 17 5x y x y 5 18 w _ w 0 ( 1_ m n ) 1 m n 6 a 5 (a ) ( x )( x _) x (x y ) 1_ (x y) 1 7 p q p q _ 19 _ 6m n 8m n 5 x 1 y 5 _ xy 1(a ) _ 9(a ) 8 n m 1. Scientific notation (standard form) Exponents provide an efficient way of writing and calculating with very large or very small numers. The need for this is especially great in science. For example, a light year (the distance that light travels in one year) is kilometres, and the mass of a single water molecule is grams. It is far more convenient and useful to write such numers in scientific notation (also called standard form). 1

13 Definition of scientific notation A positive numer N is written in scientific notation if it is expressed in the form: N a 10 k, where 1 a 10 and k is an integer In scientific notation, a light year is aout kilometres. This expression is determined y oserving that when a numer is multiplied y 10 k and k is positive, the decimal point will move k places to the right. Therefore, Knowing that when a numer is 1 decimal places multiplied y 10 k and k is negative the decimal point will move k places to the left helps us to express the mass of a water molecule as grams. This expression is equivalent to decimal places Scientific notation is also a very convenient way of indicating the numer of significant figures (digits) to which a numer has een approximated. A light year expressed to an accuracy of 1 significant figures is kilometres. However, many calculations will not require such a high degree of accuracy. For a certain calculation it may e more appropriate to have a light year approximated to significant figures, which could e written as kilometres, or more efficiently and clearly in scientific notation as kilometres. Not only is scientific notation conveniently compact, it also allows a quick comparison of the magnitude of two numers without the need to count zeros. Moreover, it enales us to use the laws of exponents to simplify otherwise unwieldy calculations. Example 5 Use scientific notation to calculate each of the following. a) ) c) a) (6. 10 )( ) ) _ _ ( 9) or 650 c) ( ) 1_ ( ) 1_ (7) 1_ (10 9 ) 1_ (7) 1_ (10 9 ) 1_ 10 or 000 Your GDC will automatically express numers in scientific notation when a large or small numer exceeds its display range. For example, if you use 1

14 1 Downloaded from your GDC to compute raised to the 6th power, the display (depending on the GDC model) will show the approximation E19 or The final digits indicate the power of 10, and we interpret the result as ( 6 is exactly ) Exercise 1. In questions 1 8, write each numer in scientific notation, rounding to significant figures Land area of Earth: square kilometres 8 Relative density of hydrogen: grams per cm In questions 9 1, write each numer in ordinary decimal notation In questions 1 16, use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to significant figures. 1 (.5 10 )( ) _ (1 10 )( ) ( 10 ) ( ) 1.5 Algeraic expressions Examples of algeraic expressions are: 5a x y 7x 8 1 (x c) y 1 a Algeraic expressions are formed y comining variales and constants using addition, sutraction, multiplication, division, exponents and radicals. Polynomials An algeraic expression that has only non-negative powers of one or more variale and contains no variale in a denominator is called a polynomial. Hint: Polynomials with one, two and three terms are called monomials, inomials and trinomials, respectively. A polynomial of degree one is called linear; degree two is called quadratic; degree three is cuic; and degree four is quartic. Quadratic equations and functions are covered in Chapter. 1 Definition of a polynomial in the variale x Given a 0, a 1, a,, a n a n 0 and n 0, n, then a polynomial in x is a sum of distinct terms in the form a n x n a n 1 x n 1 a 1 x a 0 where a 1, a,, a n are the coefficients, a 0 is the constant term, and n (the highest exponent) is the degree of the polynomial. Polynomials are added or sutracted using the properties of real numers that were discussed in Section 1.1. We do this y comining like terms terms containing the same variale(s) raised to the same power(s) and applying the distriutive property.

15 For example, x y 6x 7x y x y 7x y 6x rearranging terms so the like terms are together ( 7)x y 6x applying distriutive property: a ac ( c)a 5x y 6x no like terms remain, so polynomial is simplified Expanding and factorizing polynomials We apply the distriutive property in the other direction, i.e. a( c) a ac, in order to multiply polynomials. For example, (x )(x 5) x(x 5) (x 5) x 10x x 15 collecting like terms 10x and x x 7x 15 terms written in descending order of the exponents The process of multiplying polynomials is often referred to as expanding. Especially in the case of a polynomial eing raised to a power, the numer of terms in the resulting polynomial, after applying the distriutive property and comining like terms, has increased (expanded) compared to the original numer of terms. For example, (x ) (x )(x ) squaring a first degree (linear) inomial x(x ) (x ) x x x 9 x 6x 9 the result is a second degree (quadratic) trinomial and, (x 1) (x 1)(x 1)(x 1) cuing a first degree inomial (x 1)(x x x 1) x(x x 1) 1(x x 1) x x x x x 1 x x x 1 the result is a third degree (cuic) polynomial with four terms A pair of inomials of the form a and a are called conjugates. In most instances, the product of two inomials produces a trinomial. However, the product of a pair of conjugates produces a inomial such that oth terms are squares and the second term is negative referred to as a difference of two squares. For example, (x 5)(x 5) x(x 5) 5(x 5) multiplying two conjugates x 5x 5x 5 x 5 x 5 is a difference of two squares 15

16 1 Downloaded from The inverse (or undoing) of multiplication (expansion) is factorization. If it is helpful for us to rewrite a polynomial as a product, then we need to factorize it i.e. apply the distriutive property in the reverse direction. The previous four examples can e used to illustrate equivalent pairs of factorized and expanded polynomials. Factorized Expanded (x )(x 5) x 7x 15 (x ) x 6x 9 (x 1) x x x 1 (x 5)(x 5) x 5 Certain polynomial expansions (products) and factorizations occur so frequently you should e ale to quickly recognize and apply them. Here is a list of some of the more common ones. You can verify these identities y performing the multiplication. Common polynomial expansion and factorization patterns Expanding (x a)(x ) x (a )x a (ax )(cx d) acx (ad c)x d (a )(a ) a (a ) a a (a ) a a (a ) a a a (a ) a a a Factorizing These identities are useful patterns into which we can sustitute any numer or algeraic expression for a, or x. This allows us to efficiently find products and powers of polynomials and also to factorize many polynomials. Example 6 Find each product. a) (x )(x 7) ) (x )(x 1) c) (6x y)(6x y) d) (h 5) e) (x ) f) ( 5 )( 5 ) Hint: You should e ale to perform the middle step mentally without writing it. 16 a) This product fits the pattern (x a)(x ) x (a )x a. (x )(x 7) x ( 7)x ()( 7) x 5x 1 ) This product fits the pattern (ax )(cx d) acx (ad c)x d. (x )(x 1) 1x ( 16)x 1x 1x

17 c) This fits the pattern (a )(a ) a where the result is a difference of two squares. (5x y)(5x y) (5x ) (y) 5x 6 9y d) This fits the pattern (a ) a a. (h 5) (h) (h)(5) (5) 16h 0h 5 e) This fits the pattern (a ) a a a. (x ) (x ) (x ) () (x )() () x 6 6x 1x 8 f) The pair of expressions eing multiplied do not have a variale ut they are conjugates, so they fit the pattern (a )(a ) a. ( 5 )( 5 ) () ( 5 ) 9 ( 5) Note: The result of multiplying two irrational conjugates is a single rational numer. We will make use of this result to simplify certain fractions. Example 7 Completely factorize the following expressions. a) x 1x ) x x 15 c) x 6 9 d) y y 8y e) (x ) y f) 5x y 0xy a) x 1x (x 7x 1) factor out the greatest common factor [x ( )x ( )( )] fits the pattern (x a)(x ) x (a ) x a (x )(x ) trial and error to find 7 and ( )( ) 1 ) The terms have no common factor and the leading coefficient is not equal to one. This factorization requires a logical trial and error approach. There are eight possile factorizations. (x 1)(x 15) (x )(x 5) (x 5)(x ) (x 15)(x 1) (x 1)(x 15) (x )(x 5) (x 5)(x ) (x 15)(x 1) Testing the middle term in each, you find that the correct factorization is x x 15 (x 5)(x ) c) This inomial can e written as the difference of two squares, x 6 9 (x ) (), fitting the pattern a (a )(a ). Therefore, x 6 9 (x )(x ). 17

18 1 Downloaded from d) y y 8y y(y 8y 16) factor out the greatest common factor y(y y ) fits the pattern a a (a ) y(y ) e) Fits the difference of two squares pattern a (a )(a ) with a x and y. Therefore, (x ) y [(x ) y][(x ) y] (x y )(x y ) (f ) 5x y 0xy 5xy (x y ): although oth of the terms x and y are perfect squares, the expression x y is not a difference of squares and, hence, it cannot e factorized. The sum of two squares, a, cannot e factorized. Guidelines for factoring polynomials 1 Factor out the greatest common factor, if one exists. Determine if the polynomial, or any factors, fit any of the special polynomial patterns and factor accordingly. Any quadratic trinomial of the form ax x c will require a logical trial and error approach, if it factorizes. Most polynomials cannot e factored into a product of polynomials with integer coefficients. In fact, factoring is often difficult even when possile for polynomials with degree or higher. Nevertheless, factorizing is a powerful algeraic technique that can e applied in many situations. Algeraic fractions An algeraic fraction (or rational expression) is a quotient of two algeraic expressions or two polynomials. Given a certain algeraic fraction, we must assume that the variale can only have values such that the denominator is not zero. For example, for the algeraic fraction x x, x cannot e or. Most of the algeraic fractions that we will encounter will have numerators and denominators that are polynomials. Simplifying algeraic fractions When trying to simplify algeraic fractions, we need to completely factor the numerator and denominator and cancel any common factors. Example 8 Simplify each algeraic fraction. a) _ a a 6a 6 ) _ 1 x x x c) _ (x h) x h 18

19 a) _ a a a /(a ) 6a 6 6 /(a ) _ 1 /a 6/ _ a ) 1 x x x _ (1 x)(1 x) (x 1)(x ) ( 1 x)(1 x) _ /(x 1) (x 1) (x 1)(x ) /(x 1)(x ) _ x 1 or _ x 1 x x c) (x h) x _ x hx h x hx h _ h/ (x h) x h h h h h/ Adding and sutracting algeraic fractions Before any fractions numerical or algeraic can e added or sutracted they must e expressed with the same denominator, preferaly the least common denominator. Then the numerators can e added or sutracted according to the rule: a c _ ad c d d _ c _ ad c. d d Example 9 Perform the indicated operation and simplify. a) x 1 x ) _ a _ a a) x 1 x x 1 1 x x x 1 x x 1 x ) c) c) (x 1)(x 1) or _ x _ a _ a _ a _ a a a _ a a a a a _ 5a a _ x x x x 6 _ x _ x (x )(x ) _ x x x _ x (x )(x ) (x ) (x ) _ (x )(x ) _ x 6 x (x )(x ) _ x (x )(x ) or x x x 6 _ x x x x 6 (a ) (a ) _ (a )(a ) Hint: Although it is true that _ a a, e careful to avoid c c c an error here: _ a c a a. Also, c e sure to only cancel common factors etween numerator and denominator. It is true that ac a c (with the common factor of c cancelling) ecause ac a _ c a 1 a ; ut, in c c general, it is not true that _ a c c a. c is not a common factor of the numerator and denominator. Simplifying a compound fraction Fractional expressions with fractions in the numerator or denominator, or oth, are usually referred to as compound fractions. A compound fraction is est simplified y first simplifying oth its numerator and denominator into single fractions, and then multiplying numerator and denominator a_ a_ y the reciprocal of the denominator, i.e. c_ d_ ad _ c c_ d d d_ _ c 1 _ ad c ; therey c expressing the compound fraction as a single fraction. 19

20 1 Downloaded from Hint: Factor out the power of 1 x with the smallest exponent. Example 10 Simplify each compound fraction. 1 x h a) 1_ a_ x ) _ 1 h 1 _ a c) 1 x h a) 1 x x x(x h) _ x h _ x (x h) _ x(x h) _ x(x h) h h_ h_ 1 1 h/ x(x h) 1 1 h/ x(x h) ) c) a 1 a_ 1 _ a _ a _ a _ a _ a _ / / a _ a a _ x(1 x) (1 x) _ 1 _ 1 x _ x x h x(x h) 1 h _ x(1 x) (1 x) _ 1 (1 x) [x (1 x) 1 ] 1 x 1 x _ (1 x) [x 1 x] 1 x _ (1 x) /(1 x) _ /1 x 1 (1 x ) _ With rules for rational exponents and radicals we can do the following, ut it s not any simpler 1 _ 1 (1 x ) _ x ) _ 1 (x ) x 1 x x Rationalizing the denominator Recall Example from Section 1., where we rationalized the denominator of the numerical fractions _ and _ 7. Also recall from earlier in this 10 section that expressions of the form a and a are called conjugates and their product is a (difference of two squares). If a fraction has an _ irrational denominator of the form a c, we can change it to a rational expression ( rationalize ) y multiplying numerator and denominator y its _ conjugate a c, given that (a c )(a c ) a _ ( c ) a c. Example 11 Rationalise the denominator of each fractional expression. a) _ ) x 1 a) _ 1 5 _ _ (1 _ 5 ) 1 ( 5 ) (1 _ 5 ) 1 5 (1 5 ) 1 5 ) 1 x 1 1 _ x 1 x 1 x 1 x 1 ( x ) 1 x 1 x 1 / (1 _ 5 ) / 0

21 Exercise 1.5 In questions 1 1, expand and simplify. 1 (n )(n 5) (y )(5y ) (x 7)(x 7) (5m ) 5 (x 1) 6 (1 a )(1 a ) 7 (a )(a 1) 8 [(x ) y][(x ) y] 9 (a ) 10 (ax ) 11 (1 5 )(1 5 ) 1 (x 1)(x x 5) In questions 1 0, completely factorize the expression. 1 1x 8 1 x 6x 15 x x m m 17 x 10x y 7y 6 19 n 1n 0 0 x 0x 18x 1 a 16 y 1y 5 5n ax 6ax 9a 5 n(m 1) (m 1) 6 x (y ) 8 y 10y 96y 9 x 0x 5 0 (x ) x(x ) In questions 1 6, simplify the algeraic fraction. x 1 x n 5x 6n 6n _ a 5a 5 5 a 5 5 a 6 x x x (x h) x _ h In questions 7 6, perform the indicated operation and simplify. 7 x _ x a 1 x 1 0 _ x x 1 x 1 _ x y _ 1 x y _ x _ 5 x x x 6 _ x x y y 10 5 _ a _ 1 a 6 _ x _ 5x 6x 1 x _ y _ 5 In questions 7 50, rationalize the denominator of each fractional expression _

22 1 Downloaded from Equations and formulae One of the most famous equations in the history of mathematics, x n y n z n, is associated with Pierre Fermat ( ), a French lawyer and amateur mathematician. Writing in the margin of a French translation of Arithmetica, Fermat conjectured that the equation x n y n z n (x, y, z, n ) has no non-zero solutions for the variales x, y and z when the parameter n is greater than two. When n, the equation is equivalent to Pythagoras theorem for which there are an infinite numer of integer solutions Pythagorean triples, such as 5 and 5 1 1, and their multiples. Fermat claimed to have a proof for his conjecture ut that he could not fit it in the margin. All the other margin conjectures in Fermat s copy of Arithmetica were proven y the start of the 19th century ut this one remained unproven for over 50 years until the English mathematician Andrew Wiles proved it in 199. Equations, identities and formulae We will encounter a wide variety of equations in this course. Essentially an equation is a statement equating two algeraic expressions that may e true or false depending upon what value(s) is/are sustituted for the variale(s). The value(s) of the variale(s) that make the equation true are called the solutions or roots of the equation. All of the solutions to an equation comprise the solution set of the equation. An equation that is true for all possile values of the variale is called an identity. All of the common polynomial expansion and factorization patterns shown in Section 1.5 are identities. For example, (a ) a a is true for all values of a and. The following are also examples of identities. (x 5) (x ) x 1 (x y) xy x y Many equations are often referred to as a formula (plural: formulae) that typically contains more than one variale and, often, other symols that represent specific constants or parameters (constants that may change in value ut do not alter the properties of the expression). Formulae with which you are familiar include: A r, d rt, d (x 1 x ) (y 1 y ) and V _ r. Whereas most equations that we will encounter will have numerical solutions, we can solve a formula for a certain variale in terms of other variales sometimes referred to as changing the suject of a formula. Example 1 Solve for the indicated variale in each formula. a) a c solve for ) T 1 g solve for l a) a c c a c a If is a length then c a. ) T 1 g l_ g _ T _ l g T l T g The graph of an equation Two important characteristics of any equation are the numer of variales (unknowns) and the type of algeraic expressions it contains (e.g. polynomials, rational expressions, trigonometric, exponential, etc.). Nearly all of the equations in this course will have either one or two variales, and in this introductory chapter we will discuss only equations with algeraic expressions that are polynomials. s for equations with a single variale will consist of individual numers that can e graphed as points on a numer line. The graph of an equation is a visual representation of the

23 equation s solution set. For example, the solution set of the one-variale equation containing quadratic and linear polynomials x x 8 is x {, }. The graph of this one-variale equation is depicted (Figure 1.6) on a one-dimensional coordinate system, i.e. the real numer line Figure 1.6 The solution set. The solution set of a two-variale equation will e an ordered pair of numers. An ordered pair corresponds to a location indicated y a point on a two-dimensional coordinate system, i.e. a coordinate plane. For example, the solution set of the two-variale quadratic equation y x will e an infinite set of ordered pairs (x, y) that satisfy the equation. (Quadratic equations will e covered in detail in Chapter.) Equations of lines A one-variale linear equation in x can always e written in the form ax 0, a 0 and it will have exactly one solution, x a. An example of a two-variale linear equation in x and y is x y. The graph of this equation s solution set (an infinite set of ordered pairs) is a line. The slope m, or gradient, of a non-vertical line is defined y the formula m y _ y 1 x x _ vertical change. Because division y zero is undefined, 1 horizontal change the slope of a vertical line is undefined. Using the two points (1, 1_ ) and (, 1) we compute the slope of the line with equation x y to e m 1 ( 1_ ) _ 1 _ 1. 1 If we solve for y, we can rewrite the equation in the form y 1_ x 1. Note that the coefficient of x is the slope of the line and the constant term is the y-coordinate of the point at which the line intersects the y-axis, i.e. the y- intercept. There are several forms in which to write linear equations whose graphs are lines. Form Equation Characteristics general form ax y c 0 every line has equation in this form if oth a and 0 slope-intercept form y mx c m is the slope; (0, c) is the y-intercept y x (, ) y 6 5 (, ) 1 16 ( 5, 5) 1 0 (0, 0) 1 1 Figure 1.7 Four ordered pairs in the solution set are graphed in red. The graph of all the ordered pairs in the solution set form a curve as shown in lue. x y y (, 1) (0, 1) 1 (1, ) 7 11 (, ) Figure 1.8 The graph of x y. Tale 1.6 Forms for equations of lines. x x point-slope form y y 1 m(x x 1 ) m is the slope; (x 1, y 1 ) is a known point on the line horizontal line y c slope is zero; (0, c) is the y-intercept vertical line x c slope is undefined; unless line is y-axis, no y-intercept

24 1 Downloaded from Most prolems involving equations and graphs fall into two categories: (1) given an equation, determine its graph; and () given a graph, or some information aout it, find its equation. For lines, the first type of prolem is often est solved y using the slope-intercept form. However, for the second type of prolem, the point-slope form is usually most useful. y x 1 5 y x x 1 5 Example 1 Without using a GDC, sketch the line that is the graph of each of the following linear equations written here in general form. a) 5x y 6 0 y ) y 0 c) x 0 a) Solve for y to write the equation in slope-intercept form. 5x y 6 0 y 5x 6 y 5_ x. The line has a y-intercept of (0, ) and a slope of 5_. ) The equation y 0 is equivalent to y whose graph is a horizontal line with a y-intercept of (0, ). c) The equation x 0 is equivalent to x whose graph is a vertical line with no y-intercept; ut, it has an x-intercept of (, 0). Example 1 a) Find the equation of the line that passes through the point (, 1) and has a slope of 1. Write the equation in slope-intercept form. ) Find the linear equation in C and F knowing that when C 10 then F 50, and when C 100 then F 1. Solve for F in terms of C. a) Sustitute into the point-slope form y y 1 m(x x 1 ); x 1, y 1 1 and m 1 y y 1 m(x x 1 ) y 1 1(x ) y 1x 6 1 y 1x 5 ) The two points, ordered pairs (C, F), that are known to e on the line are (10, 50) and (100, 1). The variale C corresponds to the x variale and F corresponds to y in the definitions and forms stated aove. The slope of the line is m _ F F C C _ Choose one 5 of the points on the line, say (10, 50), and sustitute it and the slope into the point-slope form. F F 1 m(c C 1 ) F 50 9 _ 5 (C 10) F 9 _ 5 C F 9 _ 5 C

25 The slope of a line is a convenient tool for determining whether two lines are parallel or perpendicular. The two lines graphed in Figure 1.9 suggest the following property: Two distinct non-vertical lines are parallel if, and only if, their slopes are equal, m 1 m. y m 1 y x y 1 x y m 1 0 y x m x 0 x y x Figure 1.9 m Figure 1.10 The two lines graphed in Figure 1.10 suggests another property: Two nonvertical lines are perpendicular if, and only if, their slopes are negative reciprocals that is, m 1 _ m 1, which is equivalent to m 1 m 1. Distances and midpoints Recall from Section 1.1 that asolute value (modulus) is used to define the distance (always positive) etween two points on the real numer line. The distance etween the points A and B on the real numer line is B A equivalent to A B. The points A and B are the endpoints of a line segment that is denoted with the notation [AB] and the length of the line segment is denoted AB. In Figure 1.11, the distance etween A and B is AB ( ) 6. A B Figure The distance etween two general points (x 1, y 1 ) and (x, y ) on a coordinate plane can e found using the definition for distance on a numer line and Pythagoras theorem. For the points (x 1, y 1 ) and (x, y ), the horizontal distance etween them is x 1 x and the vertical distance is y 1 y. As illustrated in Figure 1.1, these distances are the lengths of two legs of a right-angled triangle whose hypotenuse is the distance etween the points. If d represents the distance etween (x 1, y 1 ) and (x, y ), then y Pythagoras theorem d x 1 x y 1 y. Because the square of any numer is positive, the asolute value is not necessary, giving us the distance formula for two-dimensional coordinates. 5 6 y y y 1 (x 1, y 1 ) x 1 Figure 1.1 x 1 x (x, y ) y 1 y (x, y 1 ) x x 5

26 1 Downloaded from The distance formula The distance d etween the two points (x 1, y 1 ) and (x, y ) in the coordinate plane is d (x 1 x ) (y 1 y ) The coordinates of the midpoint of a line segment are the average values of the corresponding coordinates of the two endpoints. The midpoint formula The midpoint of the line segment joining the points (x 1, y 1 ) and (x, y ) in the coordinate plane is ( x 1 x, y 1 y ) y 8 6 P (1, ) R (, 8) M (, ) Q (, 1) Figure x Example 15 a) Show that the points P(1, ), Q(, 1) and R(, 8)are the vertices of a right-angled triangle. ) Find the midpoint of the hypotenuse. a) The three points are plotted and the line segments joining them are drawn in Figure 1.1. Applying the distance formula, we can find the exact lengths of the three sides of the triangle. PQ (1 _ ) ( 1) _ 1 5 QR _ ( ) (1 8) _ PR _ (1 ) ( 8) _ PQ PR QR ecause ( 5 ) _ ( 5 ) _ ( 50 ). The lengths of the three sides of the triangle satisfy Pythagoras theorem, confirming that the triangle is a right-angled triangle. y (1, ) ) QR is the hypotenuse. Let the midpoint of QR e point M. Using the midpoint formula, M _ ( ), _ ( 1 8 ) ( 7, 9 ). This point is plotted in Figure d (, 10) d 1 (6, 10) Figure 1.1 The graph shows the two different points that are oth a distance of 1 from (1, ). x Example 16 Find x so that the distance etween the points (1, ) and (x, 10) is 1. d 1 (x 1) ( 10 ) 1 (x 1) ( 1) 169 x x 1 1 x x 0 (x 6)(x ) 0 x 6 0 or x 0 x 6 or x 6

27 Simultaneous equations Many prolems that we solve with algeraic techniques involve sets of equations with several variales, rather than just a single equation with one or two variales. Such a set of equations is called a set of simultaneous equations ecause we find the values for the variales that solve all of the equations simultaneously. In this section, we consider only the simplest set of simultaneous equations a pair of linear equations in two variales. We will take a rief look at three methods for solving simultaneous linear equations. They are: 1. Graphical. Sustitution. Elimination Although we will only look at pairs of linear equations in this section, it is worthwhile mentioning that the graphical and sustitution methods are effective for solving sets of equations where not all of the equations are linear, e.g. one linear and one quadratic equation. Graphical method The graph of each equation in a system of two linear equations in two unknowns is a line. The graphical interpretation of the solution of a pair of simultaneous linear equations corresponds to determining what point, or points, lies on oth lines. Two lines in a coordinate plane can only relate to one another in one of three ways: (1) intersect at exactly one point, () intersect at all points on each line (i.e. the lines are identical), or () the two lines do not intersect (i.e. the lines are parallel). These three possiilities are illustrated in Figure y y y Figure 1.15 x x x intersect at exactly one point; exactly one solution identical; coincident lines infinite solutions never intersect; parallel lines no solution Although a graphical approach to solving simultaneous linear equations provides a helpful visual picture of the numer and location of solutions, it can e tedious and inaccurate if done y hand. The graphical method is far more efficient and accurate when performed on a graphical display calculator (GDC). Example 17 Use the graphical features of a GDC to solve each pair of simultaneous equations. a) x y 6 ) 7x 5y 0 x y 10 x y 7

28 1 Downloaded from a) First, we will rewrite each equation in slope-intercept form, i.e. y mx c. This is a necessity if we use our GDC, and is also very useful for graphing y hand (manual). x y 6 y x 6 y x and x y 10 y x 10 Plot1 Plot Plot Y1=-/)X+ Y= X+10 Y= Y= Y5= Y6= Y7= CALCULATE 1:value :zero :minimum :maximum 5:intersect 6:dy/dx 7: f(x)dx Intersection X=- Y= Plot1 Plot Plot Y1=(7/5)X- Y= -X+ Y= Y= Y5= Y6= Y7= The intersection point and solution to the simultaneous equations is x and y, or (, ). If we manually graphed the two linear equations in (a) very carefully using grid paper, we may have een ale to determine the exact coordinates of the intersection point. However, using a graphical method without a GDC to solve the simultaneous equations in () would only allow us to crudely approximate the solution. Intersection X=1.666 Y= X Ans Frac 15/11 Y Ans Frac -/11 ) 7x 5y 0 5y 7x 0 y 7 5 x and x y y x The solution to the simultaneous equations is x _ and y _ 11, or _ ( 15 11, _ 11 ). The full power and efficiency of the GDC is used in this example to find the exact solution. Elimination method To solve a system using the elimination method, we try to comine the two linear equations using sums or differences in order to eliminate one of the variales. Before comining the equations we need to multiply one or oth of the equations y a suitale constant to produce coefficients for one of the variales that are equal (then sutract the equations), or that differ only in sign (then add the equations). Example 18 Use the elimination method to solve each pair of simultaneous equations. a) 5x y 9 ) x y x y 1 x y 5 8