Algebraic founations. Kick off with CAS. Algebraic skills.3 Pascal s triangle an binomial expansions.4 The binomial theorem.5 Sets of real numbers.6 Surs.7 Review
. Kick off with CAS Playing lotto Using CAS technology, calculate the following proucts. a 3 b 5 4 3 c 7 6 5 4 3 O FS 0 9 8 7 6 5 4 3 Using CAS technology, fin the symbol! an evaluate the following: a 3! b 5! c 7! O N LI N 6 E 5 PA G E 4 This symbol is calle factorial. Compare the answers to questions an. Using the factorial symbol or another metho, in how many ways can: a one number be arrange b two numbers be arrange c three numbers be arrange six numbers be arrange e nine numbers be arrange? Using the factorial symbol or another metho, answer the following. a How many -igit numbers can be forme from 6 ifferent numbers? b How many 3-igit numbers can be forme from 5 ifferent numbers? c How many 4-igit numbers can be forme from 0 ifferent numbers? In the game of lotto, how many ifferent combinations of 6 numbers can be chosen from 45 numbers? PR 3 O 0! Please refer to the Resources tab in the Prelims section of your ebookplus for a comprehensive step-by-step guie on how to use your CAS technology.
Units & AOS Topic Concept. Algebraic skills Concept summary Practice questions Worke example Algebraic skills This chapter covers some of the algebraic skills require as the founation to learning an unerstaning of Mathematical Methos. Some basic algebraic techniques will be revise an some new techniques will be introuce. review of factorisation an expansion expansion The Distributive Law a(b + c) ab + ac is funamental in expaning to remove brackets. Some simple expansions inclue: (a + b)(c + ) ac + a + bc + b (a + b) (a + b)(a + b) a + ab + b (a b) a ab + b (a + b)(a b) a b Expan (4x 3) (x )(x + ) + (x + 5)(x ) an state the coefficient of the term in x. THINK Expan each pair of brackets. Note: The first term contains a perfect square, the secon a ifference of two squares an the thir a quaratic trinomial. factorisation Some simple factors inclue: common factor ifference of two perfect squares perfect squares an factors of other quaratic trinomials. WRITE (4x 3) (x )(x + ) + (x + 5)(x ) (6x 4x + 9) (x 4) + (x x + 0x 5) Expan fully, taking care with signs. 3x 48x + 8 x + 4 + x + 9x 5 3 Collect like terms together. 33x 39x + 7 4 State the answer. Note: Rea the question again to ensure the answer given is as requeste. The expansion gives 33x 39x + 7 an the coefficient of x is 39. expan Factorise form (a + b)(c + ) is equal to factorise ac + a + bc + b Expane form 60 maths QuesT mathematical methos Vce units an
A systematic approach to factorising is isplaye in the following iagram. Common factor? ac + a a(c + ) Worke example THINK Two terms Three terms Four or more terms Difference of two squares? a b (a + b)(a b) How many terms? Quaratic trinomial? a + 3ab + b (a + b)(a + b) a + ab + b (a + b) a ab + b (a b) Grouping? ac + a + bc + b a(c + ) + b(c + ) (a + b)(c + ) Grouping terms commonly referre to as grouping an an grouping 3 an epening on the number of terms groupe together, are often use to create factors. For example, as the first three terms of a + ab + b c are a perfect square, grouping 3 an woul create a ifference of two squares expression, allowing the whole expression to be factorise. a + ab + b c (a + ab + b ) c (a + b) c This factorises to give (a + b c)(a + b + c). Factorise: a x 3 + 5x y y x b 4y x + 0x 5 a Take out the common factor. c 7(x + ) 8(x + ) + using the substitution a (x + ). WRITE a x 3 + 5x y y x x(x + 5xy y ) Factorise the quaratic trinomial. x(x 3y)(x + 4y) b The last three terms of the expression can be groupe together to form a perfect square. b 4y x + 0x 5 4y (x 0x + 5) Topic Algebraic founations 6
Use the grouping 3 an technique to create a ifference of two squares. Worke example 3 Factorising sums an ifferences of two perfect cubes Check the following by han or by using a CAS technology. Expaning (a + b)(a ab + b ) gives a 3 + b 3, the sum of two cubes; an expaning (a b)(a + ab + b ) gives a 3 b 3, the ifference of two cubes. Hence the factors of the sum an ifference of two cubes are: Factorise: a 3 + b 3 (a + b)(a ab + b ) a 3 b 3 (a b)(a + ab + b ) a x 3 7 b x 3 + 6. THINK WRITE a Express x 3 7 as a ifference of two cubes. a x 3 7 x 3 3 3 Apply the factorisation rule for the ifference of two cubes. Using a 3 b 3 (a b)(a + ab + b ) with a x, b 3, x 3 3 3 (x 3)(x + 3x + 3 ) 3 State the answer. x 3 7 (x 3)(x + 3x + 9) b Take out the common factor. b x 3 + 6 (x 3 + 8) Express x 3 + 8 as a sum of two cubes. (x 3 + 3 ) 3 Apply the factorisation rule for the sum of two cubes. 4y (x 5) (y) (x 5) 3 Factorise the ifference of two squares. [y (x 5)][y + (x 5)] 4 Remove the inner brackets to obtain the answer. (y x + 5)(y + x 5) c Substitute a (x + ) to form a quaratic trinomial in a. c 7(x + ) 8(x + ) + 7a 8a + where a (x + ) Factorise the quaratic trinomial. (7a )(a ) 3 Substitute (x + ) back in place of a (7(x + ) )((x + ) ) an simplify. 4 Remove the inner brackets an simplify to obtain the answer. (7x + 7 )(x + ) (7x + 6)(x) x(7x + 6) Using a 3 + b 3 (a + b)(a ab + b ) with a x, b, x 3 + 3 (x + )(x x + ) (x 3 + 3 ) (x + )(x x + ) 4 State the answer. x 3 + 6 (x + )(x x + 4) 6 Maths Quest MATHEMATICAL METHODS VCE Units an
Worke example 4 THINK Algebraic fractions Factorisation techniques may be use in the simplification of algebraic fractions uner the arithmetic operations of multiplication, ivision, aition an subtraction. Multiplication an ivision of algebraic fractions An algebraic fraction can be simplifie by cancelling any common factor between its numerator an its enominator. For example: ab + ac a a(b + c) a b + c For the prouct of algebraic fractions, once numerators an enominators have been factorise, any common factors can then be cancelle. The remaining numerator terms are usually left in factors, as are any remaining enominator terms. For example: a(b + c) (a + c) (b + c)(a + c) a b b Note that b is not a common factor of the numerator so it cannot be cancelle with the b in the enominator. As in arithmetic, to ivie by an algebraic fraction, multiply by its reciprocal. Simplify: a x x x 5x + 6 a Factorise both the numerator an the enominator. Note: The numerator has a common factor; the enominator is a quaratic trinomial. Cancel the common factor in the numerator an enominator. a b c a b c WRITE a b x4 x 3 + x 3 x x x x 5x + 6 x(x ) (x 3)(x ) x(x ) (x 3)(x ) 3 Write the fraction in its simplest form. x x 3 No further cancellation is possible. b Change the ivision into multiplication by replacing the ivisor by its reciprocal. b x4 x 3 + x 3 x x4 x 3 3 x + x Topic Algebraic founations 63
Factorise where possible. Note: The aim is to create common factors of both the numerator an enominator. For this reason, write (3 x) as (x 3). 3 Cancel the two sets of common factors of the numerator an enominator. 4 Multiply the remaining terms in the numerator together an the remaining terms in the enominator together. 5 State the answer. Note: The answer coul be expresse in ifferent forms, incluing as a prouct of linear factors, but this is not necessary as it oes not lea to any further simplification. Worke example 5 THINK Factorise each enominator. Aition an subtraction of algebraic fractions Factorisation an expansion techniques are often require when aing or subtracting algebraic fractions. Denominators shoul be factorise in orer to select the lowest common enominator. Express each fraction with this lowest common enominator. Simplify by expaning the terms in the numerator an collect any like terms together. Simplify 3x + 3 x + x x x. Select the lowest common enominator an express each fraction with this as its enominator. WRITE 3x + 3 x + x x x 3(x + ) (x ) + x (x + )(x ) (x ) 3(x + )(x ) Since x 4 (x ) then: x 4 x 3 3 x + x (x )(x + ) x 3 (x )(x + ) x 3 (x ) (x ) (x ) x (x )(x + ) (x 3) + x (x 3) + x 3(x + ) 3(x + )(x ) + x 3 3(x + )(x ) 64 Maths Quest MATHEMATICAL METHODS VCE Units an
(x ) 3(x + ) + 3x 3 Combine the fractions into one fraction. 3(x + )(x ) 4 Expan the terms in the numerator. Note: It is not necessary to expan the enominator terms. 5 Collect like terms in the numerator an state the answer. Note: Since there are no common factors between the numerator an the enominator, the fraction is in its simplest form. Exercise. PRactise Work without CAS Consoliate Apply the most appropriate mathematical processes an tools Algebraic skills WE Expan 3(x + ) + (7x + )(7x ) (3x + 4)(x ) an state the coefficient of the term in x. Expan ( + 3x)(x + 6)(3x )(6 x). 3 WE Factorise: a 4x 3 8x y y x b 9y x 8x 6 c 4(x 3) 3(x 3) using the substitution a (x 3). 4 Factorise x 6x + 9 xy + 3y. 5 WE3 Factorise the following. a x 3 5 b 3 + 3x 3 6 Factorise y 4 + y(x y) 3. 7 WE4 Simplify: x a + 4x x + x 8 8 Simplify x3 5 x 5 5 x 3 + 5x + 5x. 6 9 WE5 Simplify 5x 5 + x x x 6x + 5. 0 Simplify 4 x + 3 6x (x + ) x + x +. b x4 64 5 x x + 8 x 5. This exercise shoul be attempte by han rather than by using CAS technology. Expan each of the following expressions. a (x + 3) b 4a(b 3a)(b + 3a) c 0 (c + )(4c 5) (5 7y) e (3m 3 + 4n)(3m 3 4n) f (x + ) 3 Expan the following. x 4 3x 3 + 3x 3(x + )(x ) x 7 3(x + )(x ) a (g + + h) b (p + 7q) (7q p) c (x + 0)(5 + x)(0 x)(x 5) Topic Algebraic founations 65
3 Expan an simplify, an state the coefficient of the term in x. a (x 3)(x ) + (x + 5)(x ) b ( + 3x)(4 6x 5x ) (x 6)(x + 6) c (4x + 7)(4x 7)( x) (x + y)(x + + y) + (x ) e (3 x)(x + 9) 3(5x )(4 x) f x + x 4(x + x 4) 4 Factorise each of the following expressions. a x + 7x 60 b 4a 64 c bc + b + + c 5x + 7 x e 9( m) f 8x 48xy + 7y 5 Fully factorise the following. a x 3 + x 5x 50 b 00p 3 8pq c 4n + 4n + 4p 49(m + n) 8(m n) e 3(a ) + 5( a) 3 f a b a + b + (a + b ) 6 Use a substitution metho to factorise the following. a (x + 5) + (x + 5) 56 b (x + 3) 7(x + 3) 9 c 70(x + y) y(x + y) 6y x 4 8x 9 e 9(p q) + (p q ) + 4(p + q) f a a + a 7 Factorise the following. a x 3 8 b x 3 + 000 c x 3 7x 3 + 64y 3 e x 4 5x f (x ) 3 + 6 8 Fully factorise the following. a 4x 3 8y 3 b 8x 4 y 4 + xy 4a a + a + 4a c 5(x + ) 3 + 64(x 5) 3 (x y) 3 54(x + y) 3 e a 5 a 3 b + a b 3 b 5 f x 6 y 6 9 Simplify the following. a 3x 7x 0 5 9x b x3 + 4x 9x 36 x + x c (x + h)3 x 3 h e m3 m n m 4n m 3 + n 3 m + 3mn + n x 9x 3 + 3x 9x 8x x + f x3 + x x 3 + x + x + x x + x 66 maths QuesT mathematical methos Vce units an
MasTER.3 Units & AOS Topic Concept Pascal s triangle an binomial expansions Concept summary Practice questions 0 Simplify the following expressions. 4 a x + + 4 x x b c e 4 x 4 3 x + + 5 x 5 x + 6 + 4 5 x + 3 x + x 30 4y 36y + 8 + 4y 8 y 9y p q p p q q 3 p 4 q 4 f (a + 6b) 7 a 3ab + b 5 a ab b a Expan (x + 5)( x)(3x + 7). b Factorise 7(x ) 3 + 64(x + ) 3. a Simplify 3 x + 8 x + 8. b Use CAS technology to complete some of the questions in Exercise.. Pascal s triangle an binomial expansions expansions of perfect cubes The perfect square (a + b) may be expane quickly by the rule (a + b) a + ab + b. The perfect cube (a + b) 3 can also be expane by a rule. This rule is erive by expressing (a + b) 3 as the prouct of repeate factors an expaning. (a + b) 3 (a + b)(a + b)(a + b) (a + b)(a + b) (a + b)(a + ab + b ) a 3 + a b + ab + ba + ab + b 3 a 3 + 3a b + 3ab + b 3 Therefore, the rules for expaning a perfect cube are: (a + b) 3 a 3 + 3a b + 3ab + b 3 (a b) 3 a 3 3a b + 3ab b 3 The wor algebra is of arabic origin. It is erive from al-jabr, an was evelope by the mathematician Muhamma ibn Musa al-khwarizmi (c.780 850). The wor algorithm is erive from his name. features of the rule for expaning perfect cubes The powers of the first term, a, ecrease as the powers of the secon term, b, increase. The coefficients of each term in the expansion of (a + b) 3 are, 3, 3,. The coefficients of each term in the expansion of (a b) 3 are, 3, 3,. The signs alternate + + in the expansion of (a b) 3. Topic AlgebrAic founations 67
Worke example 6 THINK Expan (x 5) 3. Note: It is appropriate to use CAS technology to perform expansions such as this. WRITE Use the rule for expaning a perfect cube. (x 5) 3 Using (a b) 3 a 3 3a b + 3ab b 3, let x a an 5 b. (x 5) 3 Interactivity Pascal s triangle int-554 Pascal s triangle Although known to Chinese mathematicians many centuries earlier, the following pattern is name after the seventeenth century French mathematician Blaise Pascal. Pascal s triangle contains many fascinating patterns. Each row from row onwars begins an ens with. Each other number along a row is forme by aing the two terms to its left an right from the preceing row. Row 0 Row (x) 3 3(x) (5) + 3(x)(5) (5) 3 Simplify each term. 8x 3 3 4x 5 + 3 x 5 5 8x 3 60x + 50x 5 3 State the answer. (x 5) 3 8x 3 60x + 50x 5 4 An alternative approach to using the rule woul be to write the expression in the form (a + b) 3. (x 5) 3 (x + ( 5)) 3 Using (a + b) 3 a 3 + 3a b + 3ab + b 3, let x a an 5 b. (x 5) 3 (x + ( 5)) 3 Row Row 3 3 3 Row 4 4 6 4 The numbers in each row are calle binomial coefficients. (x) 3 + 3(x) ( 5) + 3(x)( 5) + ( 5) 3 8x 3 60x + 50x 5 68 Maths Quest MATHEMATICAL METHODS VCE Units an
Worke example 7 THINK Choose the row in Pascal s triangle which contains the require binomial coefficients. Write own the require binomial expansion. 3 State the values to substitute in place of a an b. 4 Write own the expansion. The numbers,, in row are the coefficients of the terms in the expansion of (a + b). (a + b) a + ab + b The numbers, 3, 3, in row 3 are the coefficients of the terms in the expansion of (a + b) 3. (a + b) 3 a 3 + 3a b + 3ab + b 3 Each row of Pascal s triangle contains the coefficients in the expansion of a power of (a + b). Such expansions are calle binomial expansions because of the two terms a an b in the brackets. Row n contains the coefficients in the binomial expansion (a + b) n. To expan (a + b) 4 we woul use the binomial coefficients,, 4, 6, 4,, from row 4 to obtain: (a + b) 4 a 4 + 4a 3 b + 6a b + 4ab 3 + b 4 a 4 + 4a 3 b + 6a b + 4ab 3 + b 4 Notice that the powers of a ecrease by as the powers of b increase by, with the sum of the powers of a an b always totalling 4 for each term in the expansion of (a + b) 4. For the expansion of (a b) 4 the signs woul alternate: (a b) 4 a 4 4a 3 b + 6a b 4ab 3 + b 4 By extening Pascal s triangle, higher powers of such binomial expressions can be expane. Form the rule for the expansion of (a b) 5 an hence expan (x ) 5. Note: It is appropriate to use CAS technology to perform expansions such as this. WRITE For (a b) 5, the power of the binomial is 5. Therefore the binomial coefficients are in row 5. The binomial coefficients are:, 5, 0, 0, 5, Alternate the signs: (a b) 5 a 5 5a 4 b + 0a 3 b 5 0a b 3 + 5ab 4 b 5 To expan (x ) 5, replace a by x an b by. (x ) 5 (x) 5 5(x) 4 () + 0(x) 3 () 0(x) () 3 + 5(x)() 4 () 5 5 Evaluate the coefficients an state the answer. 3x 5 5 6x 4 + 0 8x 3 0 4x + 0x 3x 5 80x 4 + 80x 3 40x + 0x (x ) 5 3x 5 80x 4 + 80x 3 40x + 0x Topic Algebraic founations 69
Exercise.3 PRactise Work without CAS Consoliate Apply the most appropriate mathematical processes an tools Pascal s triangle an binomial expansions WE6 Expan (3x ) 3. Expan a 3 + b 3 an give the coefficient of a b. 3 WE7 Form the rule for the expansion of (a b) 6 an hence expan (x ) 6. 4 Expan (3x + y) 4. 5 Expan the following. a (3x + ) 3 b ( x) 3 c (5x + y) 3 6 Select the correct statement(s). x y 3 a (x + ) 3 x 3 + 6x + x + 8 b (x + ) 3 x 3 + 3 c (x + ) 3 (x + )(x x + 4) (x + ) 3 (x + )(x + x + 4) e (x + ) 3 x 3 + 3x + 3x + 8 7 Fin the coefficient of x in the following expressions. a (x + ) 3 3x(x + ) b 3x (x + 5)(x 5) + 4(5x 3) 3 c (x )(x + )(x 3) (x ) 3 (x 3) 3 + (4 x ) 3 8 a Write own the numbers in row 7 of Pascal s triangle. b Ientify which row of Pascal s triangle contains the binomial coefficients:, 9, 36, 84, 6, 6, 84, 36, 9,. c Row 0 contains term, row contains terms. How is the number of terms relate to the row number of Pascal s triangle? 9 Copy an complete the following table by making use of Pascal s triangle. Binomial power (x + a) (x + a) 3 (x + a) 4 (x + a) 5 Expansion Number of terms in the expansion 3 Sum of inices in each term 0 Expan the following using the binomial coefficients from Pascal s triangle. a (x + 4) 5 b (x 4) 5 c (xy + ) 5 (3x 5y) 4 e (3 x ) 4 f ( + x) 6 ( x) 6 a Expan an simplify (x ) + y 4. x b Fin the term inepenent of x in the expansion of + 6. x c If the coefficient of x y in the expansion of (x + ay) 4 is 3 times the coefficient of x y 3 in the expansion of (ax y) 4, fin the value of a. Fin the coefficient of x in the expansion of ( + x)( x) 5. 70 Maths Quest MATHEMATICAL METHODS VCE Units an
Master a The sum of the binomial coefficients in row is + + 4. Form the sum of the binomial coefficients in each of rows 3 to 5. b Create a formula for the sum of the binomial coefficients of row n. c Expan ( + x) 4. In the expansion in part c, let x an comment on the result. e Using a suitably chosen value for x evaluate. 4 using the expansion in part c. 3 a Expan (x + ) 5 (x + ) 4 an hence show that (x + ) 5 (x + ) 4 x(x + ) 4. b Prove (x + ) n+ (x + ) n x(x + ) n. 4 A section of Pascal s triangle is shown. Determine the values of a, b an c. 45 a b 65 330 0 c 5 Pascal s triangle can be written as: 3 3 4 6 4 5 0 0 5 a Describe the pattern in the secon column. b What woul be the sixth entry in the thir column? c Describe the pattern of the terms in the thir column by forming a rule for the nth entry. What woul be the rule for fining the nth entry of the fourth column? 6 Expan ( + x + x ) 4 an hence, using a suitably chosen value for x, evaluate 0.9 4. The Yang Hui (Pascal s) triangle as epicte in 303 in a work by the Chinese mathematician Chu Shih-chieh Topic Algebraic founations 7
.4 Units & AOS Topic Concept 3 The binomial theorem Concept summary Practice questions Interactivity The binomial theorem int-555 Worke example 8 The binomial theorem Note: The binomial theorem is not part of the Stuy Design but is inclue here to enhance unerstaning. Pascal s triangle is useful for expaning small powers of binomial terms. However, to obtain the coefficients require for expansions of higher powers, the triangle nees to be extensively extene. The binomial theorem provies the way aroun this limitation by proviing a rule for the expansion of (x + y) n. Before this theorem can be presente, some notation nees to be introuce. Factorial notation In this an later chapters, calculations such as 7 6 5 4 3 will be encountere. Such expressions can be written in shorthan as 7! an are rea as 7 factorial. There is a factorial key on most calculators, but it is avisable to remember some small factorials by heart. Definition n! n (n ) (n ) 3 for any natural number n. It is also necessary to efine 0!. 7! is equal to 5040. It can also be expresse in terms of other factorials such as: 7! 7 6 5 4 3 7! 7 6 5 4 3 or 7 6! 7 6 5! This is useful when working with fractions containing factorials. For example: 7! 6! 7 6! 5! 6! 7! 5! 7 6 5! or 7 4 By writing the larger factorial in terms of the smaller factorial, the fractions were simplifie. Factorial notation is just an abbreviation so factorials cannot be combine arithmetically. For example, 3!!!. This is verifie by evaluating 3!!. Evaluate 5! 3! + 50! 49! 3!! 3 6 4 THINK WRITE Expan the two smaller factorials. 5! 3! + 50! 49! 5 4 3 3 + 50! 49! 7 Maths Quest MATHEMATICAL METHODS VCE Units an
To simplify the fraction, write the larger factorial in terms of the smaller factorial. Formula for binomial coefficients Each of the terms in the rows of Pascal s triangle can be expresse using factorial notation. For example, row 3 contains the coefficients, 3, 3,. 3! These can be written as 0! 3!, 3!!!, 3!!!, 3! 3! 0!. (Remember that 0! was efine to equal.) The coefficients in row 5 (, 5, 0, 0, 5, ) can be written as: 5! 0! 5!, 5!! 4!, 5!! 3!, 5! 3!!, 5! 4!!, 5! 5! 0! 4! The thir term of row 4 woul equal an so on.!! n! The (r + )th term of row n woul equal. This is normally written using n r! (n r)! the notations n C r or r. These expressions for the binomial coefficients are referre to as combinatoric coefficients. They occur frequently in other branches of mathematics incluing probability theory. Blaise Pascal is regare as the father of probability an it coul be argue he is best remembere for his work in this fiel. n r 5 4 3 3 + 50 49! 49! 0 6 + 50 64 3 Calculate the answer. 0 6 + n! r!(n r)! n C r, where r n an r, n are non-negative whole numbers. Pascal s triangle with combinatoric coefficients Pascal s triangle can now be expresse using this notation: 0 Row 0 0 Row 0 Row Row 3 Row 4 3 4 3 4 3 4 3 4 0 0 0 3 3 50 49! 49! 4 4 Topic Algebraic founations 73
Worke example 9 THINK Apply the formula. Binomial expansions can be expresse using this notation for each of the binomial coefficients. The expansion (a + b) 3 3 0 a3 + 3 a b + 3 ab + 3 3 b3. Note the following patterns: n 0 n (the start an en of each row of Pascal s triangle) n n n n (the secon from the start an the secon from the en of each n n n row) an r n r. While most calculators have a n C r key to assist with the evaluation of the coefficients, n n the formula for or n C r shoul be known. Some values of can be committe to memory. r r Evaluate 8 3. Write the largest factorial in terms of the next largest factorial an simplify. WRITE n n! r r!(n r)! Let n 8 an r 3. 8 8! 3 3!(8 3)! 8! 3!5! 8 7 6 5! 3!5! 8 7 6 3! 3 Calculate the answer. 8 7 6 3 8 7 56 Binomial theorem The binomial coefficients in row n of Pascal s triangle can be expresse as n 0, n, n, n n an hence the expansion of (x + y)n can be forme. The binomial theorem gives the rule for the expansion of (x + y) n as: (x + y) n n 0 xn + n xn y + n xn y +... + n r xn r y r +... + n n yn 74 Maths Quest MATHEMATICAL METHODS VCE Units an
Since n 0 n n this formula becomes: (x + y) n x n + n xn y + n xn y + + n r x n r y r + + y n Worke example 0 THINK Write out the expansion using the binomial theorem. Note: There shoul be 5 terms in the expansion. Evaluate the binomial coefficients. 3 Complete the calculations an state the answer. Features of the binomial theorem formula for the expansion of (x + y) n There are (n + ) terms. In each successive term the powers of x ecrease by as the powers of y increase by. For each term, the powers of x an y a up to n. For the expansion of (x y) n the signs alternate + + +... with every even term assigne the sign an every o term assigne the + sign. Use the binomial theorem to expan (3x + ) 4. WRITE (3x + ) 4 (3x) 4 + 4 (3x)3 () + 4 (3x) () + 4 3 (3x) ()3 + () 4 (3x) 4 + 4 (3x) 3 () + 6 (3x) () + 4 (3x)() 3 + () 4 8x 4 + 4 7x 3 + 6 9x 4 + 4 3x 8 + 6 (3x + ) 4 8x 4 + 6x 3 + 6x + 96x + 6 Using the binomial theorem The binomial theorem is very useful for expaning (x + y) n. However, for powers n 7 the calculations can become quite teious. If a particular term is of interest then, rather than expan the expression completely to obtain the esire term, an alternative option is to form an expression for the general term of the expansion. The general term of the binomial theorem Consier the terms of the expansion: (x + y) n x n + n xn y + n xn y + + n r xn r y r + + y n Term : t n 0 xn y 0 Term : t n xn y Term 3: t 3 n xn y Topic Algebraic founations 75
Following the pattern gives: Term (r + ): t r+ n r xn r y r Worke example THINK For the expansion of (x + y) n, the general term is t r+ n r xn r y r. For the expansion of (x y) n, the general term coul be expresse as t r+ n r xn r ( y) r. The general-term formula enables a particular term to be evaluate without the nee to carry out the full expansion. As there are (n + ) terms in the expansion, if the mile term is sought there will be two mile terms if n is o an one mile term if n is even. Fin the fifth term in the expansion of WRITE x y 3 State the general term formula of the expansion. x y 3 The (r + )th term is t r+ n n r x y r 3 Since the power of the binomial is 9, n 9. t r+ 9 9 r x y r r 3 Choose the value of r for the require term. For the fifth term, t 5 : r + 5 r 4 9 t 5 9 4 9 4 3 Evaluate to obtain the require term. 6 x5 3 y4 8 7x5 y 4 44 x 9 x. 9 4 5 y 3 y 3 4 4 r. Ientifying a term in the binomial expansion The general term can also be use to etermine which term has a specifie property such as the term inepenent of x or the term containing a particular power of x. 76 Maths Quest MATHEMATICAL METHODS VCE Units an
Worke example Ientify which term in the expansion of (8 3x ) woul contain x 8 an express the coefficient of x 8 as a prouct of its prime factors. THINK Exercise.4 PRactise Work without CAS The binomial theorem WE8 Evaluate 6! + 4! 0! 9!. Simplify n! (n )!. WRITE Write own the general term for this expansion. (8 3x ) Rearrange the expression for the general term by grouping the numerical parts together an the algebraic parts together. 3 Fin the value of r require to form the given power of x. The general term: t r+ t r+ r (8) r ( 3) r (x ) r r (8) r ( 3) r x r For x 8, r 8 so r 4. r (8) r ( 3x ) r 4 Ientify which term is require. Hence it is the fifth term which contains x 8. 5 Obtain an expression for this term. With r 4, t 5 4 (8) 4 ( 3) 4 x 8 4 (8)8 ( 3) 4 x 8 6 State the require coefficient. The coefficient of x 8 is 4 (8)8 ( 3) 4. 7 Express the coefficient in terms of prime numbers. 4 (8)8 ( 3) 4 0 9 ( 3 ) 8 3 4 4 3 5 9 4 3 4 5 3 4 3 4 5 3 6 4 8 State the answer. Therefore the coefficient of x 8 is 5 3 6 4. 3 WE9 Evaluate 7 4. 4 Fin an algebraic expression for n an use this to evaluate. Topic Algebraic founations 77
5 WE0 Use the binomial theorem to expan (x + 3) 5. 6 Use the binomial theorem to expan (x ) 7. 7 WE Fin the fourth term in the expansion of 8 Fin the mile term in the expansion of x + y x 3 y 0. 7. Consoliate Apply the most appropriate mathematical processes an tools 9 WE Ientify which term in the expansion of (4 + 3x 3 ) 8 woul contain x 5 an express the coefficient of x 5 as a prouct of its prime factors. 0 Fin the term inepenent of x in the expansion of x + x Evaluate the following. a 6! b 4! +! c 7 6 5! 6! 3! e 0! 9! f (4! + 3!) Evaluate the following. a 6! b 4! c 49! 4! 43! 50! 69!! + 0! 70!! 0! 3 Simplify the following. a (n + ) n! b (n )(n )(n 3)! c n! (n 3)! (n )! (n + )! e n! (n + )! 4 Evaluate the following. a 5 0 C 3 5 Simplify the following. b e 5 3 7 0 (n )! (n + )! f n3 n n (n )! (n + )! n a n 3 n + n 6 Expan the following. b e n n 3 n + n 3 c f c f 6. 3 0 n + 3 n n + a (x + ) 5 b ( x) 5 c (x + 3y) 6 7 x + e x x 8 3 f (x + ) 0 78 Maths Quest MATHEMATICAL METHODS VCE Units an
Master 7 Obtain each of the following terms. a The fourth term in the expansion of (5x + ) 6 b The tenth term in the expansion of ( + x) c The sixth term in the expansion of (x + 3) 0 The thir term in the expansion of (3x ) 6 e The mile term(s) in the expansion of (x 5) 6 f The mile term(s) in the expansion of (x + y) 7 8 a Specify the term which contains x 4 in the expansion of (x + 3). b Obtain the coefficient of x 6 in the expansion of ( x ) 9. c Express the coefficient of x 5 in the expansion of (3 + 4x) as a prouct of its prime factors. Calculate the coefficient of x x in the expansion of. x e Fin the term inepenent of x in the expansion of x + 0. x 3 f Fin the term inepenent of x in the expansion of x + x 8 6 x x 9 a Determine the value of a so that the coefficients of the fourth an the fifth terms in the expansion of ( + ax) 0 are equal. b If the coefficient of x in the expansion of ( + x + x ) 4 is equal to the coefficient of x in the expansion of ( + x) n, fin the value of n. 0 a Use the expansion of ( + x) 0 with suitably chosen x to show that 0 0 + 0 + 0 + + 0 0 0 an interpret this result for Pascal s triangle. b Show that n + n r r + n r an interpret this result for Pascal s triangle. Evaluate the following using CAS technology. a 5! b 5 0 a Solve for n: b Solve for r: n 770 r 0 6. I m very well acquainte, too, with matters mathematical; I unerstan equations, both the simple an quaratical; About Binomial Theorem I am teeming with a lot o news, With many cheerful facts about the square of the hypotenuse! Source: Verse of I am the very moel of a moern major general from Pirates of Penzance by Gilbert an Sullivan. Topic Algebraic founations 79
.5 Units & AOS Topic Concept 4 Real numbers Concept summary Practice questions Interactivity Sets int-556 Sets of real numbers The concept of numbers in counting an the introuction of symbols for numbers marke the beginning of major intellectual evelopment in the mins of the early humans. Every civilisation appears to have evelope a system for counting using written or spoken symbols for a few, or more, numbers. Over time, technologies were evise to assist in counting an computational techniques, an from these counting machines the computer was evelope. Over the course of history, ifferent categories of numbers have evolve which collectively form the real number system. Real numbers are all the numbers which are positive or zero or negative. Before further escribing an classifying the real number system, a review of some mathematical notation is given. Set notation A set is a collection of objects, these objects being referre to as the elements of the set. The elements may be liste as, for example, the set A,, 3, 4, 5 an the set B, 3, 5. The statement A means is an element of set A, an the statement B means oes not belong to, or is not an element of, set B. Since every element in set B, 3, 5 is also an element of set A,, 3, 4, 5, B is a subset of set A. This is written as B A. However we woul write A B since A is not a subset of B. The union of the sets A an B contains the elements which are either in A or in B or in both. Element shoul not be counte twice. The intersection of the sets A an B contains the elements which must be in both A an B. This is written as A B an woul be the same as the set B for this example. The exclusion notation A \ B exclues, or removes, any element of B from A. This leaves a set with the elements, 4. Sets may be given a escription as, for example, set C x : < x < 0. The set C is rea as C is the set of numbers x such that x is between an 0. The set of numbers not in set C is calle the complement of C an given the symbol C. The escription of this set coul be written as C x : x or x 0. A set an its complement cannot intersect. This is written as C C where is a symbol for empty set. Such sets are calle isjoint sets. There will be ongoing use of set notation throughout the coming chapters. Classification of numbers While counting numbers are sufficient to solve equations such as + x 3, they are not sufficient to solve, for example, 3 + x where negative numbers are neee, nor 3x where fractions are neee. 80 Maths Quest MATHEMATICAL METHODS VCE Units an
The following sequence of subsets of the real number system, while logical, oes not necessarily reflect the historical orer in which the real number system was establishe. For example, fractions were establishe long before the existence of negative numbers was accepte. Natural numbers are the positive whole numbers or counting numbers. The set of natural numbers is N,, 3,. The positive an negative whole numbers, together with the number zero, are calle integers. The set of integers is Z,, 0,,, 3,. The symbol Z is erive from the German wor zahl for number. Rational numbers are those which can be expresse as quotients in the form p q, where q 0, an p an q are integers which have no common factors other than. The symbol for the set of rational numbers is Q (for quotients). Rational numbers inclue finite an recurring ecimals as well as fractions an integers. For example: 9 75, 0.75 8 8 00 3 4, 0.3 0.3333 3, an 5 5 are rational. Natural numbers an integers are subsets of the set of rational numbers with N Z Q. Irrational numbers are numbers which are not rational; they cannot be expresse in fraction form as the ratio of two integers. Irrational numbers inclue numbers such as an π. The set of irrational numbers is enote by the symbol Q using the complement symbol for not. Q Q as the rational an irrational sets o not intersect. The irrational numbers are further classifie into the algebraic irrationals an the non-algebraic ones known as transcenental numbers. Algebraic irrationals are those which, like rational numbers, can be solutions to an equation with integer coefficients, while transcenental numbers cannot. For example, π is transcenental while is algebraic since it is a solution of the equation x 0. R The union of the set of rational an irrational numbers Qʹ Q forms the set of real numbers R. Z Hence R Q Q. This is isplaye in the iagram N showing the subsets of the real numbers. The set of all real numbers forms a number line continuum on which all of the positive or zero or negative numbers are place. Hence R R 0 R +. R Zero R + The sets which forme the builing blocks of the real number system have been efine, enabling the real number system to be viewe as the following hierarchy. 0 Real numbers Rationals Irrationals Integers Fractions Algebraic Transcenental Topic Algebraic founations 8
Worke example 3 Expressions an symbols that o not represent real numbers It is important to recognise that the following are not numbers. The symbol for infinity may suggest this is a number but that is not so. We can speak of numbers getting larger an larger an approaching infinity, but infinity is a concept, not an actual number. Any expression of the form a oes not represent a number since ivision by zero is 0 not possible. If a 0, the expression 0 is sai to be ineterminate. It is not efine 0 as a number. To illustrate the secon point, consier 3 0. Suppose 3 ivie by 0 is possible an results in a number we shall call n. 3 0 n 3 0 n 3 0 The conclusion is nonsensical so 3 is not efine. 0 However, if we try the same process for zero ivie by zero, we obtain: 0 0 n 0 0 n 0 0 While the conclusion hols, it is not possible to etermine a value for n, so 0 0 is ineterminate. It is beyon the Mathematical Methos course, but there are numbers that are not elements of the set of real numbers. For example, the square roots of negative numbers, such as, are unreal, but these square roots are numbers. They belong to the set of complex numbers. These numbers are very important in higher levels of mathematics. a Classify each of the following numbers as an element of a subset of the real numbers. i 3 5 ii 7 iii 6 3 iv 9 b Which of the following are correct statements? i 5 Z iii R R + R ii Z N THINK WRITE a i Fractions are rational numbers. a i 3 5 Q ii Surs are irrational numbers. ii 7 Q 8 Maths Quest MATHEMATICAL METHODS VCE Units an
iii Evaluate the number using the correct orer of operations. iv Evaluate the square root. iv 9 ±3 9 Z Interval notation Interval notation provies an alternative an often convenient way of escribing certain sets of numbers. Close interval [a, b] x : a x b is the set of real numbers that lie between a an b, incluing the enpoints, a an b. The inclusion of the enpoints is inicate by the use of the square brackets [ ]. This is illustrate on a number line using close circles at the enpoints. a b Open interval (a, b) x : a < x < b is the set of real numbers that lie between a an b, not incluing the enpoints a an b. The exclusion of the enpoints is inicate by the use of the roun brackets ( ). This is illustrate on a number line using open circles at the enpoints. Half-open intervals Half-open intervals have only one enpoint inclue. [a, b) x : a x < b iii 6 3 6 6 0 (6 3) Z. An alternative answer is (6 3) Q. b i Z is the set of integers. b i 5 Z is a correct statement since 5 is an integer. ii N is the set of natural numbers. ii Z N is incorrect since N Z. iii This is the union of R, the set of negative real numbers, an R +, the set of positive real numbers. iii R R + R is incorrect since R inclues the number zero which is neither positive nor negative. a b (a, b] x : a < x b a b a b Topic Algebraic founations 83
Interval notation can be use for infinite intervals using the symbol for infinity with an open en. For example, the set of real numbers, R, is the same as the interval (, ). Worke example 4 THINK a b a Illustrate the following on a number line an express in alternative notation. i (, ] ii x : x iii,, 3, 4. b Use interval notation to escribe the sets of numbers shown on the following number lines. i 0 3 4 5 6 i Describe the given interval. Note: The roun bracket inicates not inclue an a square bracket inicates inclue. Write the set in alternative notation. WRITE a ii 3 4 5 6 7 i (, ] is the interval representing the set of numbers between an, close at, open at. 0 An alternative notation for the set is (, ] x : < x. ii Describe the given set. ii x : x is the set of all numbers greater than or equal to. This is an infinite interval which has no right-han enpoint. Write the set in alternative notation. iii Describe the given set. Note: This set oes not contain all numbers between the beginning an en of an interval. Write the set in alternative notation. i Describe the set using interval notation with appropriate brackets. ii Describe the set as the union of the two isjoint intervals. b iii 0 3 4 5 6 An alternative notation is x : x,, 3, 4 is a set of iscrete elements. Alternative notations coul be,, 3, 4 x : x 4, x N, or,, 3, 4 [, 4] N. [, ). i The set of numbers lie between 3 an 5, with both enpoints exclue. The set is escribe as (3, 5). Describe the same set by consiering the interval that has been exclue from R. ii The left branch is (, 3] an the right branch is [5, ). The set of numbers is the union of these two. It can be escribe as (, 3] [5, ). Alternatively, the iagram can be interprete as showing what remains after the set (3, 5) is exclue from the set R. An alternative escription is R \ (3, 5). 84 Maths Quest MATHEMATICAL METHODS VCE Units an
Exercise.5 PRactise Work without CAS Consoliate Apply the most appropriate mathematical processes an tools Sets of real numbers WE3 a Classify each of the following numbers as an element of a subset of the real numbers. i 6 ii 7 iii (6 ) 3 iv 0.5 b Which of the following are correct statements? i 7 N ii Q N iii Q Q R x 5 For what value(s) of x woul be unefine? (x + )(x 3) 3 WE4 a Illustrate the following on a number line an express in alternative notation. i [, ) ii x : x < iii,, 0,, b Use interval notation to escribe the sets of numbers shown on the following number lines. i 0 3 4 5 6 ii 3 4 5 6 7 4 Write R \ x : < x 4 as the union of two sets expresse in interval notation. 5 Which of the following oes not represent a real number? a 3 4 b 0 c 6 3 4 7 5π (8 4) + 3 e 8 4 6 Explain why each of the following statements is false an then rewrite it as a correct statement. 4 a 6 + 5 Q b 9 Z c R + x : x 0.5 Q 7 Select the irrational numbers from the following set of numbers.,,, π,, π 8 State whether the following are true or false. a R R b N R + c Z N R Q Z Z e Q Z R \ Q f Z \ N Z 9 Determine any values of x for which the following woul be unefine. a b x + x + 8 c x + 5 x (x + 3)(5 x) 0 Use interval notation to escribe the intervals shown on the following number lines. a c 0 3 4 5 3 4 5 6 7 b 4 x 4x 3 4 5 6 7 8 9 0 3 4 5 6 Topic Algebraic founations 85
Master Express the following in interval notation. a x : 4 < x 8 b x : x > 3 c x : x 0 x : x 0 Show the following intervals on a number line. a [ 5, 5) b (4, ) c [ 3, 7] ( 3, 7] e (, 3] f (, ) 3 Illustrate the following on a number line. a R \ [, ] b (, ) (, ) c [ 4, ) (0, 4) [ 4, ) (0, 4) e, 0, f R \ 0 4 Use an alternative form of notation to escribe the following sets. a x : < x < 6, x Z b R \ (, 5] c R (, 4) [, ) 5 Determine which of the following are rational an which are irrational numbers. a 75 b 75600 c 0.343434... 6 The ancient Egyptians evise the formula A 64 8 for calculating the area A of a circle of iameter. Use this formula to fin a rational approximation for π an evaluate it to 9 ecimal places. Is it a better approximation than 7? PAGE.6 Surs Units & AOS Topic Concept 5 Surs Concept summary Practice questionsonline Orering surs PROOFS A sur is an nth root, n x. surs are irrational numbers, an cannot be expresse in the quotient form p. Hence, surs have neither a finite nor a recurring ecimal form. q Any ecimal value obtaine from a calculator is just an approximation. All surs have raical signs, such as square roots or cube roots, but not all numbers with raical signs are surs. For surs, the roots cannot be evaluate exactly. Hence, 6 is a sur. 5 is not a sur since 5 is a perfect square, 5 5, which is rational. Surs are real numbers an therefore have a position on the number line. To estimate the position of 6, we can place it between two rational numbers by placing 6 between its closest perfect squares. 86 Maths Quest MATHEMATICAL METHODS VCE Units an
Units & AOS Topic Concept 6 Simplifying surs Concept summary Practice questions Worke example 5 THINK 4 < 6 < 9 4 < 6 < 9 < 6 < 3 So 6 lies between an 3, closer to, since 6 lies closer to 4 than to 9. Checking with a calculator, 6.4495. Note that the symbol always gives a positive number, so the negative sur 6 woul lie on the number line between 3 an at the approximate position.4495. To orer the sizes of two surs such as 3 5 an 5 3, express each as an entire sur. 3 5 9 5 5 3 5 3 9 5 an 5 3 45 75 Since 45 < 75 it follows that 3 5 < 5 3. Surs in simplest form Surs are sai to be in simplest form when the number uner the square root sign contains no perfect square factors. This means that 3 5 is the simplest form of 45 an 5 3 is the simplest form of 75. If the raical sign is a cube root then the simplest form has no perfect cube factors uner the cube root. To express 8 in its simplest form, fin perfect square factors of 8. a Express each number entirely as a square root. 8 64 64 8 8 8 a Express 6, 4 3, 5, 7 with its elements in increasing orer. b Express in simplest form i 56 ii 5a b assuming a > 0. WRITE a 6, 4 3, 5, 7 6 36 7, 4 3 6 3 48, 5 4 5 0 an 7 49. Orer the terms. 0 < 48 < 49 < 7 In increasing orer, the set of numbers is 5, 4 3, 7, 6. Topic Algebraic founations 87
b i Fin a perfect square factor of the number uner the square root sign. ii Fin any perfect square factors of the number uner the square root sign. b i 56 4 4 4 4 4 ii 5a b 4 9 7a b Express the square root terms as proucts an simplify where possible. 3 Try to fin the largest perfect square factor for greater efficiency. 4 9 7 a b 3 7 a b a 7b Alternatively, recognising that 5 is 36 7, 5a b 36 7a b 36 7 a b 6 7 a b a 7b Operations with surs As surs are real numbers, they obey the usual laws for aition an subtraction of like terms an the laws of multiplication an ivision. Aition an subtraction a c + b c (a + b) c a c b c (a b) c Suric expressions such as + 3 cannot be expresse in any simpler form as an 3 are unlike surs. Like surs have the same number uner the square root sign. Expressing surs in simplest form enables any like surs to be recognise. Multiplication an ivision c (c) a c b (ab) (c) c c Note that c c because c c c c c. 88 Maths Quest MATHEMATICAL METHODS VCE Units an
Worke example 6 Simplify the following. a 3 5 + 7 + 6 5 3 b 3 98 4 7 + 5 c 4 3 6 5 THINK WRITE a Collect like surs together an simplify. a 3 5 + 7 + 6 5 3 3 5 + 6 5 + 7 3 9 5 + 4 b Write each sur in simplest form. b 3 98 4 7 + 5 3 49 4 36 + 5 5 3 7 4 6 + 5 5 4 + 0 5 Collect like surs together. ( 4 ) + 0 5 3 + 0 5 c Multiply the rational numbers together an multiply the surs together. c 4 3 6 5 (4 6) (3 5) 4 45 Write the sur in its simplest form. 4 9 5 4 3 5 7 5 Expansions Expansions of brackets containing surs are carrie out using the istributive law in the same way as algebraic expansions. a( b + c) ab + ac ( a + b)( c + ) ac + a + bc + b The perfect squares formula for binomial expansions involving surs becomes: a ± b a ± ab + b, since: ( a ± b) ( a) ± a b + ( b) a ± ab + b The ifference of two squares formula becomes a + b a b a b since: ( a + b)( a b) ( a) ( b) a b The binomial theorem can be use to expan higher powers of binomial expressions containing surs. Topic Algebraic founations 89
Worke example 7 Expan an simplify the following. a 3 (4 5 6) b ( 3 5 )(4 5 + 4) c (3 3 + 5) ( 7 + 3 )( 7 3 ) THINK a Use the istributive law to expan, then simplify each term. WRITE a 3 4 5 6 4 5 5 4 3 4 5 3 4 30 3 b Expan as for algebraic terms. b 3 5 4 5 + 4 8 5 + 4 0 0 5 8 Simplify where possible. 8 5 + 4 0 0 5 4 7 8 5 + 4 0 0 5 7 8 5 + 4 0 0 0 7 There are no like surs so no further simplification is possible. c Use the rule for expaning a perfect square. c 3 3 + 5 Simplify each term remembering that a a an collect any like terms together. Use the rule for expaning a ifference of two squares. (3 3) + (3 3) ( 5) + ( 5) 9 3 + 3 3 5 + 4 5 7 + 5 + 0 47 + 5 7 + 3 7 3 ( 7) (3 ) 7 9 7 8 Rationalising enominators It is usually esirable to express any fraction whose enominator contains surs as a fraction with a enominator containing only a rational number. This oes not necessarily mean the rational enominator form of the fraction is simpler, but it can provie a form which allows for easier manipulation an it can enable like surs to be recognise in a suric expression. The process of obtaining a rational number on the enominator is calle rationalising the enominator. There are ifferent methos for rationalising enominators, epening on how many terms there are in the enominator. 90 Maths Quest MATHEMATICAL METHODS VCE Units an
Rationalising monomial enominators a Consier, where a, b, c Q. This fraction has a monomial enominator since its b c enominator contains the one term, b c. In orer to rationalise the enominator of this fraction, we use the fact that c c c, a rational number. Multiply both the numerator an the enominator by c. As this is equivalent to multiplying by, the value of the fraction is not altere. a b c a b c c c a c b( c c) a c bc a By this process b c a c an the enominator, bc, is now rational. bc Once the enominator has been rationalise, it may be possible to simplify the expression if, for example, any common factor exists between the rationals in the numerator an enominator. Rationalising binomial enominators a + b an a b are calle conjugate surs. Multiplying a pair of conjugate surs always results in a rational number since a + b a b a b. This fact is use to rationalise binomial enominators. Consier where a, b Q. This fraction has a binomial enominator since a + b its enominator is the aition of two terms. To rationalise the enominator, multiply both the numerator an the enominator by a b, the conjugate of the sur in the enominator. This is equivalent to multiplying by, so the value of the fraction is unaltere; however, it creates a ifference of two squares on the enominator. a + b a + b a b a b a b ( a + b)( a b) a b a b By this process we have rational number. a + b a b where the enominator, a b, is a a b Topic Algebraic founations 9
Worke example 8 a Express the following with a rational enominator. i ii 5 3 3 0 5 3 4 5 b Simplify 5 4 + 3 8. THINK a 6 c Express with a rational enominator. 5 3 3 Given p 3, calculate, expressing the answer with a rational p enominator. i The enominator is monomial. Multiply both numerator an enominator by the sur part of the monomial term. Multiply the numerator terms together an multiply the enominator terms together. 3 Cancel the common factor between the numerator an enominator. ii The enominator is monomial. Multiply both numerator an enominator by the sur part of the monomial term. WRITE a i 5 3 5 3 3 3 3 5 3 3 5 4 3 5 5 4 3 5 ii 5 3 3 0 4 5 5 3 3 0 4 5 5 5 3 3 0 4 5 5 5 3 50 0 Simplify the surs, where possible. 5 5 3 5 0 3 Take out the common factor in the numerator since it can be cancelle as a factor of the enominator. 5 5 5 0 5 5 3 0 5 3 4 5 5 9 Maths Quest MATHEMATICAL METHODS VCE Units an
b Rationalise any enominators containing surs an simplify all terms in orer to ientify any like surs that can be collecte together. c The enominator is binomial. Multiply both numerator an enominator by the conjugate of the binomial sur containe in the enominator. Expan the ifference of two squares in the enominator. Note: This expansion shoul always result in a rational number. 3 Cancel the common factor between the numerator an the enominator. Note: The numerator coul be expane but there is no further simplification to gain by oing so. b 5 4 + 3 8 5 4 + 3 3 5 4 + 9 5 + 9 c The conjugate of 5 3 3 is 5 3 + 3 6 5 3 3 6 5 3 3 5 3 + 3 5 3 + 3 6 5 3 + 3 5 3 3 5 3 + 3 6 5 3 + 3 5 3 3 6 5 3 + 3 5 3 9 6 5 3 + 3 57 6(5 3 + 3 ) 57 9 (5 3 + 3 ) or 0 3 + 6 9 9 Substitute the given value an simplify. Given p 3, p (3 ) (9 6 + ) 8 6 Factorise the enominator so that the binomial sur is simpler. 6 3 Topic Algebraic founations 93
3 Multiply numerator an enominator by the conjugate of the binomial sur containe in the enominator. 6 3 3 + 3 + 3 + 6 3 3 + 4 Expan the ifference of two squares an simplify. Exercise.6 PRactise Work without CAS Surs WE5 a Express 3 3, 4 5, 5, 5 with its elements in increasing orer. b Express the following in simplest form. i 84 ii 08ab assuming b > 0 Express 3 384 in simplest form. 3 WE6 Simplify the following. a 5 4 7 7 5 + 3 7 b 3 48 4 7 + 3 3 c 3 5 7 5 4 Simplify 5 80 + 0 + 43 + 5. 5 WE7 Expan an simplify the following. a 3 4 5 + 5 3 b 3 8 5 5 c 4 3 5 3 5 3 5 + 6 a Expan + 3. 3 + 6 (3) 3 + 6(9 ) 3 + 6 7 3 + 4 b If + 6 3 + 6 6 a + b 3, a, b N, fin the values of a an b. 7 WE8 a Express the following with a rational enominator. 6 i ii 3 5 + 7 5 7 3 b Simplify 4 6 + 6 5 4. 0 c Express with a rational enominator. 4 3 + 3 Given p 4 3 +, calculate, expressing the answer with a rational p enominator. 94 Maths Quest MATHEMATICAL METHODS VCE Units an
Consoliate Apply the most appropriate mathematical processes an tools 8 a Simplify 3 3 + 3 by first rationalising each enominator. 3 + 3 b Show that 3 + + 3 + is rational by first placing each fraction on a 3 common enominator. 9 Select the surs from the following set of real numbers. 8, 900, 4 9,.44, 03, π, 3 7, 3 36 0 Express the following as entire surs. a 4 5 b 3 6 c 9 7 4 3 3 e ab c f m 3 n Express each of the following in simplifie form. a 75 b 5 48 c 000 3 88 e 7 f 3 54 Simplify the following. a 3 7 + 8 3 + 7 9 3 b 0 6 + 4 6 8 c 3 50 8 8 45 + 5 e 6 + 7 5 + 4 4 8 0 f 7 43 + 8 3 6 3 Carry out the following operations an express answers in simplest form. a 4 5 7 b 0 6 8 0 c 3 8 5 8 7 4 7 47 6 e f 5 3 4 5 3 6 + 3 7 0 4 Expan an simplify the following. a 3 5 7 6 b 5 3 7 3 3 + 6 c 0 3 6 3 5 + 6 3 + 5 3 + 4 7 e 5 3 6 3 + 3 0 f 3 x 3 y 3 x + 3 xy + 3 y 5 Expan the following. a + 3 b 3 6 3 c 7 5 3 5 + 3 5 3 e 0 3 5 0 + 3 5 f 3 + + 3 + 6 Express the following in simplest form with rational enominators. a 3 4 3 6 + 5 + b e 0 + 5 0 3 c 8 f 3 3 + 3 + Topic Algebraic founations 95
Master 7 Express the following as a single fraction in simplest form. a 4 5 6 + 3 6 0 b ( 0 + 9 8) 3 5 3 c 3( + 3) + 3 e 6 4 7 4 + 9 7 3 3 + + 3 + 3 f ( 3) + 3 + 3 4 3 8 a If x 3 0, calculate the value of the following. i x + x ii x 4 3x 5 5 b If y 7 +, calculate the value of the following. 7 i y ii y y c Determine the values of m an n for which each of the following is a correct statement. i 7 + 3 7 3 m 7 + n 3 ii ( + 3) 4 7 3 ( + 3) m + n The real numbers x an x are a pair of conjugates. If x b + i state x ii calculate the sum x + x iii calculate the prouct x x. b 4ac : a 9 A triangle has vertices at the points A(, ), B( 5, 0) an C( 0, 5). a Calculate the lengths of each sie of the triangle in simplest sur form. b An approximation attributable to the Babylonians is that a ± b a ± b a. Use this formula to calculate approximate values for the lengths of each sie of the triangle. c Calculate from the sur form the length of the longest sie to ecimal place. 0 A rectangular lawn has imensions 6 + 3 + m by 3 + m. Hew agrees to mow the lawn for the householer. a Calculate the exact area of the lawn. b If the householer receive change of $3.35 from $50, what was the cost per square metre that Hew charge for mowing the lawn? Aristotle was probably the first to prove was what we call irrational an what he calle incommensurable. Plato, an ancient Greek philosopher, claime his teacher Theoorus of Cyrene, builing on Aristotle s approach, was the first to prove the irrationality of the non-perfect squares from 3 to 7. The work of Theoorus no longer exists. 96 Maths Quest MATHEMATICAL METHODS VCE Units an
ONLINE ONLY.7 Review www.jacplus.com.au The Maths Quest Review is available in a customisable format for you to emonstrate your knowlege of this topic. The Review contains: short-answer questions proviing you with the opportunity to emonstrate the skills you have evelope to efficiently answer questions without the use of CAS technology Multiple-choice questions proviing you with the opportunity to practise answering questions using CAS technology ONLINE ONLY Activities To access ebookplus activities, log on to www.jacplus.com.au Interactivities A comprehensive set of relevant interactivities to bring ifficult mathematical concepts to life can be foun in the Resources section of your ebookplus. Extene-response questions proviing you with the opportunity to practise exam-style questions. a summary of the key points covere in this topic is also available as a igital ocument. REVIEW QUESTIONS Downloa the Review questions ocument from the links foun in the Resources section of your ebookplus. stuyon is an interactive an highly visual online tool that helps you to clearly ientify strengths an weaknesses prior to your exams. You can then confiently target areas of greatest nee, enabling you to achieve your best results. Units & Algebraic founations Sit topic test Topic AlgebrAic founations 97