144 Maths Quest 9 for Victoria

Transcription

1 Expanding 5 Belinda works for an advertising company that produces billboard advertising. The cost of a billboard is based on the area of the sign and is $50 per square metre. If we increase the length of the sign by m and the height of the sign by m, can you write a rule for the increase in the cost of the billboard? This chapter shows you how to manipulate algebraic terms and expressions to place them in the most useful form.

2 Maths Quest 9 for Victoria Expanding single brackets In the previous chapter on introductory algebra, we learned that algebra is a type of language. In this chapter we can take the idea further and look at some of the advanced features of this language. At first we will rely on using common numerical examples to illustrate the techniques, but after a while, when your confidence has increased, the techniques will become easier to understand using algebra only. What is expanding? Consider the English word: won t. It really stands for will not. In going from won t to will not we have expanded the word, but the meaning remains unchanged. It is the same with expanding in algebra; we go from a more compact form (won t) to an expanded form (will not). Consider the following example with numbers. ( + 5) How can we find its value? We know from our work on order of operations in chapter that we do brackets first, so that: ( + 5) (9) Now the brackets mean multiply, so: ( + 5) (9) 7 Consider now an alternative way of expanding the original expression, temporarily ignoring order of operations. ( + 5) () + (5) This may seem unusual, or even incorrect, but it isn t. (It is not!) ( + 5) () + (5) You can try this with any numbers you like, but the result is the same. This expansion is valid and correct. Expansion means to multiply everything inside the brackets by what is directly outside the brackets.

3 Chapter 5 Expanding 5 WORKED Example Expand the following expressions. a 5( + ) b 5(x + ) c 5(x y) d a(x y) a Write the expression. a 5( + ) Expand the brackets. 5() + 5() Multiply out the brackets Check that the result is valid by simplifying the brackets in the original expression first. 5( + ) 5(7) 5, so it is indeed valid. b Write the expression. b 5(x + ) Expand the brackets. 5(x) + 5() Multiply out the brackets. 5x + 5 c Write the expression. c 5(x y) Expand the brackets. Multiply out the brackets. (Remember that a positive term multiplied by a negative term makes a negative term.) 5(x) + 5( y) 5x 5y d Write the expression. d a(x y) Expand the brackets. a(x) a( y) Multiply out the brackets. (Remember that a negative term multiplied by a negative term makes a positive term.) ax + ay Note: It doesn t matter what is immediately outside the brackets. It may be a number or a pronumeral or both. The following expansions are a little more complex. WORKED Example Expand each of the following. a 5x(6y 7z) b y(x + w) c x(x + y) a Write the expression. a 5x(6y 7z) Expand the brackets. Multiply out the brackets. (Multiply number parts and pronumeral parts separately and write pronumerals for each term in alphabetical order.) 5x(6y) + 5x( 7z) 0xy 5xz b Write the expression. b y(x + w) Expand the brackets. y(x) y(w) Multiply out the brackets. 8xy wy c Write the expression. c x(x + y) Expand the brackets. x(x) + x(y) Multiply out the brackets. (Remember that x multiplied by itself gives x.) x + xy

4 6 Maths Quest 9 for Victoria Expanding and collecting like terms With more complicated expansions, like terms may need to be collected after the expansion of the bracketed part. Remember that like terms contain the same pronumeral parts. You first expand the brackets, then collect the like terms. WORKED Example Expand and simplify by collecting like terms. a (x ) + 5 b x(y ) + 5x c x(y z) + 5x d 7x + 6(y x) a Write the expression. a (x ) + 5 Expand the brackets. (x) + ( ) + 5 Multiply out the brackets. x Collect any like terms. x b Write the expression. b x(y ) + 5x Expand the brackets. x(y) + x( ) + 5x Multiply out the brackets. xy x + 5x Collect any like terms. xy + x c Write the expression. c x(y z) + 5x Expand the brackets. x(y) + x( z) + 5x Multiply out the brackets. xy xz + 5x Collect any like terms. There are no like terms. d Write the expression. d 7x + 6(y x) Expand the brackets. 7x + 6(y) + 6( x) Multiply out the brackets. 7x + 6y x Collect any like terms. 5x + 6y remember remember. Expansion means to multiply everything inside the brackets by what is directly outside the brackets.. After expanding brackets, simplify by collecting any like terms. EXCEL GC Spreadsheet Mathcad program Expanding single brackets Expanding single brackets Expanding WORKED Example WORKED Example 5A Expanding single brackets Expand the following expressions. a (x + ) b (x + ) c 5(m + ) d (p + 5) e (x + ) f 7(x ) g (y + 6) h 5(a + ) i (p ) j (x ) k (x + ) l (x ) m (b ) n 8(m ) o 6(5m ) p (9p 5) Expand each of the following. a x(x + ) b y(y + ) c a(a + 5) d c(c + ) e x( + x) f y(5 + y) g m(7 m) h q(8 q) i x(y + ) j 5p(q + ) k y(x + ) l 0p(q + 9) m b( a) n 7m(5 n) o 6a(5 a) p x(7 x)

5 Chapter 5 Expanding 7 WORKED ORKED Example Expand and simplify by collecting like terms. a (p ) + b 5(x 5) + 8 c 7(p + ) d (p ) e 6x(x ) x f m(m + 5) m g x(p + ) 5 h y(y ) + 7 i p(p ) + 5p j 5(x y) y x k m(m 5) + m l p(p q) + pq m 7a(5 b) + 5a ab n c(d c) cd 5c o 6p + (p + 5) p 5 9m + (m ) SkillSHEET 5. Oops! Any errors? Here are 6 expressions that someone has simplified. Have any errors been made? a 5(x ) 5x b x + 6x 9 + x c (x) x d 8x x 5 e (x 7) x f x(x + 5) x + 5x Which expressions have been simplified correctly? Explain why someone might make the errors you have found. Correct any errors you find by rewriting the expression on the right-hand side of the equals sign. Choose values for x and evaluate the left- and right-hand sides to check whether they are now equivalent. The Bagels game In the game of Bagels, a player is to determine a -digit number (no digit repeated) by making educated guesses. After each guess, a clue is given about the guess. Here are the clues. bagels: no digit correct pico: one digit is correct but in the wrong position fermi: one digit is correct and in the correct position In each of the problems below, a number of guesses have been made with the clue for each guess shown to its right. From the given set of guesses and clues, determine the -digit number. a bagels b 908 bagels 56 pico pico 789 pico 87 pico fermi 075 pico fermi 56 fermi 087 pico 7 pico pico?????? Now try this game with a partner. One person is to decide on a -digit number and provide clues to the other person who is guessing what the -digit number is.

6 8 Maths Quest 9 for Victoria What is a metre? e? Expand the brackets in the expressions given to find the puzzle s answer code. e ae 6e e e ae a e ae 5e 0 8 e a 6 e ae ae e 8 e 5e 0 6a a e ae 6a 6e e 8 e ae e a ae e ae e + e e + e e ae a a ae e 6e e a + ae a a a 6 6e e a + ae 8 e 5e 0 e ae a e ae 6e e 6a ae e 6 a 8 e a 6 ae ae a a a ae 6e e a + ae 8 e ae e a a ae ae 6e e a + ae ae + 6e a a e + e 8 e 6e e a a 6e e a + ae 8 e 8 e 5a 0ae a a 6a 6e e a a ae ae (a ) ( a) 5(e ) (e ) (a + ) a( + e) e( a) e(e + ) a (e a) e ( a + ) a( a) e(a + ) 5a( e) ae(e ) ae(a e) e( e) a( a + )

7 Chapter 5 Expanding 9 Expanding two brackets When expanding an expression that contains two (or more) brackets, the steps are the same as before. Step Expand each bracket (working from left to right). Step Collect any like terms. WORKED Example Expand and simplify the following expressions. a 5(x + y) + 6(x y) b 5x(y ) + y(x + ) c 7y(x y) + y (x + 5) d 5xy( + y) + 6x(y + x) a Write the expression. a 5(x + y) + 6(x y) Expand each bracket. 5(x) + 5(y) + 6(x) + 6( y) Multiply out the brackets. 5x + 0y + 6x 8y Collect any like terms. x 8y b Write the expression. b 5x(y ) + y(x + ) Expand each bracket. 5x(y) 5x( ) + y(x) + y() Multiply out the brackets. 5xy + 0x + xy + y Collect any like terms. xy + 0x + y c Write the expression. c 7y(x y) + y (x + 5) Expand each bracket. 7y(x) + 7y( y) + y (x) + y (5) Multiply out the brackets. 7xy y + xy + 5y Collect any like terms. 7xy 9y + xy d Write the expression. d 5xy( + y) + 6x(y + x) Expand each bracket. 5xy() 5xy(y) + 6x(y) + 6x(x) Multiply out the brackets. 5xy 0xy + 6xy + x Collect any like terms. xy 0xy + x As you can see, there is really no difference between the questions in this section and the previous section; just a little more complex bookkeeping is required. Be careful when collecting like terms which may only appear to be like. For example, x y and xy are not like terms at all. remember remember To expand an expression:. expand each bracket (working from left to right). collect any like terms.

8 50 Maths Quest 9 for Victoria 5B Expanding two brackets GC SkillSHEET Mathcad program 5. WORKED Example a Expanding two brackets Expanding WORKED Example b, c, d Expand and simplify the following expressions. a (x + y) + (x y) b (p + q) + (p q) c 7(a + b) + (a + b) d 5(c + d) + (c + d) e (m + n) + (m n) f (x + y) + (x y) g (x + y) + (5x + y) h 5(p + q) + (p + q) i 6(a b) 5(a b) j 5(x y) (x y) k (p q) (p q) l (c d) 5(c d) m 7(x y) (x y) n 5(p q) (p q) o (a b) (a + b) p (c + d) (c + d) Expand and simplify the following expressions. a a(b + ) + b(a ) b x(y + ) + y(x ) c c(d ) + c(d + 5) d p(q 5) + p(q + ) e c(d ) + c(d 5) f 7a(b ) b(a + ) g m(n + ) m(n + ) h c(d 5) + c(d 8) i m(m + ) (m + 5) j 5c(d ) (c + cd) k a(5a + b) + b(b a) l c(c 6d) + d(d c) m 6m(m ) (m + ) n p(p ) + (5p ) o 7x(5 x) + 6(x ) p y(5y ) (y + ) multiple choice a What is the equivalent of (a + b) + (a b)? A 5a + 6b B 7a + b C 5(a + b) D 7a + 8b E a b b What is the equivalent of (x y) (x 5y)? A x + y B x y C x + y D x + 7y E x + 0y c What is the equivalent of m(n + ) + m(n )? A m + n 8 B 5mn + m C 5mn + 0m D 5mn + 6m E 6mn 6m MATHS MATHS QUEST C H A L LL E N G G E E Find the next terms of this algebraic sequence. x + y, x + y, x + y, 5x + 5y, 8x + 9y, x + y,. Mind-reading tricks often use algebra as a base. Try the following mind-reading trick. Use algebra to explain why the trick works. Double the number of the month in which you were born. Subtract 6 from your answer. Subtract 0 from your result, then multiply by 0. Finally, add the day of the month in which you were born to your answer. The number you end up with shows the month and day you were born. For example, if you were born on June 7, your answer will be 67. Try this trick on another person. Read the person s mind by stating the month and day they were born from the number they tell you at the end of the calculation.

9 Chapter 5 Expanding 5 Expanding pairs of brackets In the previous section we expanded expressions with two brackets which were separated by a + or sign, such as 5(x + y) + 6(x y). In this section we begin to look at expressions where there are two brackets being multiplied together, such as (x + y) (x y). These require a more careful analysis and technique. Let us again refer to a numerical example: ( + 6)(7 ) The traditional approach, using order of operations from chapter, results in: ( + 6)(7 ) (8)() Consider the alternative approach. First multiply the by the second bracket and then the 6 by the second bracket. ( + 6)(7 ) (7 ) + 6(7 ) (7) + ( ) + 6(7) + 6( ) Again, we end up with identical results, no matter which method is used. It may appear unnecessarily long for numeric examples (and, indeed, it is) but it works well for algebraic expressions. When multiplying expressions within brackets, multiply each term in the first bracket by each term in the second bracket. WORKED Example 5 Expand and simplify each of the following expressions. a (6 5)(5 + ) b (x 5)(x + ) c (x + )(x + ) d (x + )(x + ) a Write the expression. a (6 5)(5 + ) Expand by multiplying each of the 6(5 + ) 5(5 + ) terms in the first bracket by each of the terms in the second bracket. Expand each of the remaining brackets. 6(5) + 6() 5(5) 5() Simplify Check the results using the order of operations method. (6 5)(5 + ) ()(8) 8 b Write the expression. b (x 5)(x + ) Expand by multiplying each of the x(x + ) 5(x + ) terms in the first bracket by each of the terms in the second bracket. Expand each of the remaining brackets. (x)(x) + x() 5(x) 5() x + x 5x 5 Collect like terms. x x 5 Continued over page

10 5 Maths Quest 9 for Victoria c Write the expression. c (x + )(x + ) Expand by multiplying each of the terms in the first x(x + ) + (x + ) bracket by each of the terms in the second bracket. Expand each of the remaining brackets. x(x) + x() + (x) + () x + x + x + 6 Collect like terms. x + 5x + 6 d Write the expression. d (x + )(x + ) Expand by multiplying each of the terms in the first x(x + ) + (x + ) bracket by each of the terms in the second bracket. Expand each of the remaining brackets. x(x) + x() + (x) + () x + 6x + x + 6 Collect like terms. x + 0x + 6 Note: The last step in each question (collection of like terms), is most important, and often forgotten. Without completing this step, the expansion is not fully correct. An alternative method Part of the problem in expanding pairs of brackets is a bookkeeping one keeping track of which terms have been multiplied by which. An alternative approach uses a diagram to keep track of the various multiplication operations. Use a diagrammatic technique to expand (x + y)(x 5z). Write the expression and add curved lines connecting each term, according to the pattern shown. Number each of the curved lines. (x + y)(x 5z) 5 WORKED Example 6 Perform each of the multiplications, in order of the numbers on the lines. Write the expression and its expansion by placing the terms on a single line. Collect like terms if necessary (none in this example). : x(x) 8x : x( 5z) 0xz : y(x) xy : y( 5z) 5yz (x + y)(x 5z) 8x 0xz + xy 5yz There is nothing magical or special about this method, but it forces you to keep track of the multiplications required in the expansion. This method is often given the name FOIL, where the letters stand for: O First multiply the first term in each bracket. F Outer multiply the outer terms. (a + b) (c + d) Inner multiply the inner terms. I L Last multiply the last term of each bracket.

11 remember remember Chapter 5 Expanding 5. When multiplying expressions within pairs of brackets, multiply each term in the first bracket by each term in the second bracket, then collect the like terms.. You can use a diagrammatic method (or FOIL) to help you keep track of which terms are to be multiplied together. 5C Expanding pairs of brackets WORKED Example 5 WORKED Example 6 Expand and simplify each of the following expressions. a (a + )(a + ) b (x + )(x + ) c (y + )(y + ) d (m + )(m + 5) e (b + )(b + ) f (p + )(p + ) g (a )(a + ) h (x )(x + 5) i (m + )(m ) j (y + 5)(y ) k (y 6)(y + ) l (x )(x + ) m (x )(x ) n (p )(p ) o (x )(x ) Use a diagrammatic technique to expand the following. a (a + )(a + ) b (m + )(m + ) c (6x + )(x + ) d (c 6)(c 7) e (7 t)(5 t) f ( x)(9 x) g ( + t)(5 t) h (7 5x)( x) i (5x )(5x ) Expand and simplify each of the following. a (x + y)(z + ) b (p + q)(r + ) c (x + y)(z + ) d (p + q)(r + ) e (a + b)(a + b) f (c + d)(c d) g (x + y)(x y) h (p q)(p + q) i (y + z)(x + z) j (a + b)(b + c) k (p q)( r) l (7c d)(d 5) m (x y)(x y) n (p q)(p r) o (5 j)(k + ) multiple choice a The equivalent of (x + 7)(x ) is: A x + 5x B x + 5 C x 5x D x + 5x + E x 5x + b What is the equivalent of ( y)(7 + y)? A 8 y B 8 y + y C 8 y y D y E 8 + y y c The equivalent of (p + )(p 5) is: A p 5 B p p 5 C p 9p 5 D p 6p 5 E p + 9p 5 Expanding pairs of brackets Expanding GC Expanding 00 WorkSHEET Mathcad program GAMEtime 5. MATHS MATHS QUEST C H A L LL E N G G E E Xavier left for school in the morning. One quarter of the way to school, he passed a post office. The clock on the outside of the post office showed 7. am. Halfway to school, he passed a convenience store. The time shown there was 7.5 am. If Xavier continues walking at the same speed, at what time will he get to school? The human heart beats about 0 5 times each day. Approximately how many times does the heart beat in an 80-year lifetime? There are people trying out for a tennis team. Five of them are girls. What percentage of the possible doubles teams could be mixed double teams?

12 5 Maths Quest 9 for Victoria What has area got to do with expanding? Draw a square of side length x. x What is the area of this square? Consider the rectangle at right which has x a length units longer and a width units wider than the square you have just drawn. Notice that it has been divided up into regions. Find the area of each of the regions. What is the total area of the rectangle? 5 Write an expression for the length of the rectangle. 6 Write an expression for the width of the rectangle. 7 Using the relationship, area length width and your answers to parts 5 and 6, write an expression for the area of the rectangle. 8 Relate your answers to parts and 7. What have you noticed? 9 Show, using a diagram and areas of regions, how to obtain the expanded expression for (x + 5)(x + 7). 0 Show, using a diagram and areas of regions, how to obtain the expanded expression for (a + b)(c + d). Challenge: Show, using a diagram and areas of regions, how to obtain the expanded expression for (x )(x + ). Expand 5(x + ). Expand z( 7z). Expand (p 7q) + p 5q and simplify by collecting like terms. Expand and simplify 5(a + b) + (a + b). 5 Expand and simplify m(n + ) + n(m ). 6 multiple choice If K(a b) (a + 0b) a b, then the missing number is: A B C D E 5 7 Expand and simplify (a + )(a + ). 8 Expand and simplify (p 7)(6p ). 9 Expand and simplify ( j)(k + ). 0 True or false? (x + )(x + 0) x + x + 0

13 Chapter 5 Expanding 55 Expansion patterns Although the techniques learned in the previous section are perfectly adequate for all expansions of pairs of brackets, there are some special cases where the expansion is particularly simple and can be done very quickly if you recognise the pattern. After comparing the result with that obtained using previous methods, perhaps you will adopt these short cuts. Difference of two squares rule The first pattern we will examine is obtained as a result of expanding a pair of brackets to produce a difference of two squares. That is, we produce two terms which are perfect squares (can be expressed as a number and/or a pronumeral squared) where one term is subtracted from the other. Consider expanding (x + )(x ). (x + )(x ) x(x ) + (x ) x(x) + x( ) + (x) + ( ) x x + x 9 x 9 Notice how the middle terms, x + x cancel each other out. This is the key to the pattern and will always happen. (Can you prove this?) Note: The terms left over are the squares of each of the original terms. In other words, (x + )(x ) x. Notice the pattern of terms in the pair of brackets which produce the difference of two squares. Here are some more examples. (x + 5)(x 5) (x + )(x ) (x + h)(x h) (x + 7)(x 7) x 5 x x h (x) 7 x 5 x 6 x 9 Therefore, when we recognise this pattern we merely have to write the squares of each of the two terms and place a minus sign between them. WORKED Example 7 (a + b)(a b) a b Use the difference of two squares rule, if possible, to expand and simplify each of the following. a (x + 8)(x 8) b (6 x)(6 + x) c (x )(x + ) d (5 + x)(5 x) a Write the expression. a (x + 8)(x 8) Check that the expression can be written as the difference of two squares by comparing it with (a + b)(a b). It can. Write the answer as the difference of two squares using the formula x 8 x 6 (a + b)(a b) a b, where a x and b 8. Continued over page

14 56 Maths Quest 9 for Victoria b Write the expression. b (6 x)(6 + x) Check that the difference of two squares rule can be used. It can, because (6 x)(6 + x) means the same as (6 + x)(6 x). Write the answer as the difference of two squares using the formula 6 x 6 x (a + b)(a b) a b, where a 6 and b x. c Write the expression. c (x )(x + ) Check that the difference of two squares rule can be used. It can. Write the answer as the difference of two squares using the formula (x) x 9 (a + b)(a b) a b, where a x and b. d Write the expression. d (5 + x)(5 x) Check that the difference of two squares rule can be used. It can. Write the answer as the difference of two squares using the formula (a + b)(a b) a b, where a 5 and b x. 5 (x) 5 9x Expanding identical brackets (perfect squares) The next pattern worth examining is the expansion of identical brackets. That is, each bracket is the same, such as (x + )(x + ), which can be written as (x + ). Again, we can express this using the two symbols a and b. So a perfect square could be written in the form (a + b)(a + b). Let us try this with a set of numbers first; using the pair of brackets ( + )( + ) which equals 9. ( + )( + ) ( + ) + ( + ) () + () + () + () (Can you see a pattern here?) Now let s try it with pronumerals. (x + )(x + ) x(x + ) + (x + ) x(x) + x() + (x) + () x + x + x + 9 x + 6x + 9 There is a pattern in these expansions, fairly similar to the one we just learned. (a + b)(a + b) a + ab + b There is an additional result that occurs when there is a minus sign instead of a plus sign: (a b)(a b) a ab + b The difference between the two patterns is quite small: the minus sign in the brackets results in a single minus sign in the middle term, ab. This pattern can also be described in words. Square the first term, add the square of the last term, then add (or subtract) twice their product.

15 Chapter 5 Expanding 57 WORKED Example Use the identical brackets (perfect squares) technique to expand and simplify the following. a (x + )(x + ) b (x ) c (x + 5) d (x 5y) 8 a Write the expression. a (x + )(x + ) Recognise the pattern of identical (x)(x) x brackets: square the first term. Square the last term. ()() Add (because of the + sign in the (x)() x bracket) twice the product. 5 Apply the formula: (a + b)(a + b) a + ab + b. (x + )(x + ) x + x + b Write the expression and express it b (x ) (x )(x ) as a pair of brackets. Recognise the pattern of identical (x)(x) x brackets: square the first term. Square the last term. ( )( ) Subtract (because of the sign in the (x)() x bracket) twice the product. 5 Apply the formula: (a b)(a b) a ab + b. (x ) x x + c Write the expression and express it as a pair of brackets. c (x + 5) (x + 5)(x + 5) Recognise the pattern of identical (x)(x) x brackets: square the first term. Square the last term. (5)(5) 5 Add twice the product. (x)(5) 0x 5 Apply the formula: (x + 5) x + 0x + 5 (a + b)(a + b) a + ab + b. d Write the expression and express it as a pair of brackets. d (x 5y) (x 5y)(x 5y) Recognise the pattern of identical (x)(x) 6x brackets: square the first term. Square the last term. ( 5y)( 5y) 5y Subtract twice the product. (x)(5y) 0xy 5 Apply the formula: (a b)(a b) a ab + b. (x 5y) 6x 0xy + 5y remember remember. The difference of two squares rule is: (a + b)(a b) a b.. The identical brackets (perfect squares) rules are: (a + b)(a + b) a + ab + b (a b)(a b) a ab + b.

16 58 Maths Quest 9 for Victoria 5D Expansion patterns EXCEL EXCEL Spreadsheet Mathcad WORKED Example 7a, b Expanding (ax + b)(ax b) Expansion patterns Spreadsheet Expanding (ax + b) WORKED Example 7c, d WORKED Example 8a, b WORKED ORKED Example 8c, d Use the difference of two squares rule, if possible, to expand and simplify each of the following. a (x + )(x ) b (y + )(y ) c (m + 5)(m 5) d (a + 7)(a 7) e (x + 6)(x 6) f (p )(p + ) g (a + 0)(a 0) h (m )(m + ) i (p q)(p + q) Use the difference of two squares rule, if possible, to expand and simplify each of the following. a (x + )(x ) b (y )(y + ) c (5d )(5d + ) d (7c + )(7c ) e ( + p)( p) f ( 9x)( + 9x) g (5 a)(5 + a) h ( + 0y)( 0y) i (b 5c)(b + 5c) Use the identical brackets (perfect squares) rules to expand and simplify each of the following. a (x + )(x + ) b (a + )(a + ) c (b + 7)(b + 7) d (c + 9)(c + 9) e (m + ) f (n + 0) g (x 6) h (y 5) i (9 c) j (8 + e) k (x + y) l (u v) Use the identical brackets (perfect squares) rules to expand and simplify each of the following. a (a + ) b (x + ) c (m 5) d (x ) e (5a ) f (7p + ) g (9x + ) h (c 6) i ( + a) j (5 + p) k ( 5x) l (7 a) m (9x y) n (8x y) o (9x y) p (7x y) Using expanding formulas to square large numbers Can you evaluate 997 without a calculator and in less than 90 seconds? We would be able to evaluate this using long multiplication, but it would take a fair amount of time and effort. Mathematicians are always looking for quick and simple ways of solving problems. What if we consider the expanding formula which produces the difference of two squares? (a + b)(a b) a b Adding b to both sides gives (a + b)(a b) + b a b + b. Simplifying and swapping sides gives a (a + b)(a b) + b. We can use this new formula and the fact that multiplying by 000 is an easy operation to evaluate 997. If a 997, what should we make the value of b become so that (a + b) equals 000? Substitute these a and b values into the formula to evaluate 997. Try this method to evaluate the following. a 995 b 990 c 99 d 999 e Can you use the expanding formulas (a + b) a + ab + b or (a b) a ab + b to evaluate 997? Explain your method for this. 5 List three examples of your own and show how you were able to evaluate them using the method from part.

17 Chapter 5 Expanding 59 More complicated expansions Although we have covered many expansion problems and patterns, these represent only a small proportion of the possible algebraic expressions that can be expanded. Nevertheless, the techniques we have learned so far are very useful more complicated expansions just require more bookkeeping. The most important of these bookkeeping functions is the collection of like terms after expansion. Expanding more than two brackets There are several possible combinations, such as expanding brackets, brackets, and so on. WORKED Example Expand and simplify each of the following expressions. a (x + )(x + ) + (x ) b (x )(x + ) (x )(x + ) c (x + )(x ) (x + )(x + ) d (x + )(x ) + (x ) a Write the expression. a (x + )(x + ) + (x ) Expand and simplify the first pair of brackets. (x + )(x + ) x + x + x + x + 7x + Expand the last bracket. (x ) x 8 Add the two results. (x + )(x + ) + (x ) x + 7x + + x 8 5 Collect like terms. x + x + b Write the expression. b (x )(x + ) (x )(x + ) Expand and simplify the first pair of brackets. Expand and simplify the second pair of brackets. Subtract all of the second result from the first result. (So, place the second result in a bracket.) Remember that (x + x ) (x + x ). (x )(x + ) x + x x 6 x + x 6 (x )(x + ) x + x x x + x (x )(x + ) (x )(x + ) x + x 6 (x + x ) 5 Collect like terms. x + x 6 x x + c Write the expression. c (x + )(x ) (x + )(x + ) 5 9 Expand the first pair of brackets. It is a (x + )(x ) difference of two squares expansion. x Expand the second pair of brackets. It is an (x + )(x + ) identical bracket expansion. x + x + Subtract the two results. (x + )(x ) (x + )(x + ) x (x + x + ) Collect like terms. x x x x 8 Continued over page

18 60 Maths Quest 9 for Victoria d Write the expression. d (x + )(x ) + (x ) Expand and simplify the first pair of brackets. Then multiply by the coefficient of outside the pair. (x + )(x ) (x x + x ) (x x ) x x Expand the second pair of brackets. It is an identical bracket expansion. (x ) x x + Add the two results. (x + )(x ) + (x ) x x + (x x + ) 5 Collect like terms. x x + x x + x 6x 0 remember remember. Brackets or pairs of brackets that are added or subtracted must be expanded separately.. Always collect any like terms following an expansion. 5E More complicated expansions GC SkillSHEET SkillSHEET Mathcad program More complicated expansions Expanding WORKED Example 9 Expand and simplify each of the following expressions. (x + )(x + 5) + (x + )(x + ) (x + )(x + ) + (x + )(x + ) (x + 5)(x + ) + (x + )(x + ) (x + )(x + ) + (x + )(x + ) 5 (p )(p + 5) + (p + )(p 6) 6 (a + )(a ) + (a )(a ) 7 (p )(p + ) + (p + )(p 5) 8 (x )(x + ) + (x )(x + 0) 9 (y )(y + ) + (y )(y + ) 0 (d + 7)(d + ) + (d + )(d ) (x + )(x + ) + (x )(x ) (y + 6)(y ) + (y )(y ) (x + ) + (x 5)(x ) (y ) + (y + )(y ) 5 (p + )(p + 7) + (p ) 6 (m 6)(m ) + (m + 5) 7 (x + )(x + 5) (x + )(x + 5) 8 (x + 5)(x + ) (x + )(x + ) 9 (x + )(x + ) (x + )(x + ) 0 (m )(m + ) (m + )(m ) (b + )(b 6) (b )(b + ) (y )(y 5) (y + )(y + 6) (p )(p + ) (p )(p ) (x + 7)(x + ) (x )(x ) 5 (c )(c ) (c + 6)(c + 7) 6 ( f 7)( f + ) ( f + )( f + 5) 7 (m + ) (m + )(m ) 8 (a 6) (a )(a ) 9 (p )(p + ) (p + ) 0 (x + 5)(x ) (x )

19 Chapter 5 Expanding 6 Why does the giraffe have a long neck? Expand and simplify the expressions to find the puzzle s answer. (x )(x + ) (x + 5)(x ) (x + ) + (x ) x(x ) + x(6 x) (x )(x + ) + 6 x (x ) + (x + ) (x + 5)(x ) + (x + ) (x )(x + ) x (x ) 5(x + ) (x ) (x + )(x ) + (x + )(x ) (x 7)(x + 5) + 5(x + ) (x + x 6) 5(x + x + ) (x + x) (x )(x + 6) + (x ) (x + ) + (x ) (x + 7) (x + 5)(x 8) (x 6)(x 7) + (7x ) (x )(x ) 6 7( x) + 5(x ) (x 6) (x 8) (x + x 7) (x + 7x 7) 5x( x) (x + ) (x x + ) (x + ) (x 5)(x + 5) (x + )(x ) 5(x 7) + (x x + 8) (x ) (x + ) ( + x)( x) + (x + )(x + ) (x + x ) + 5(x ) (x + )(x ) (x )(x + ) x x + x x x 5x + 8x x + x + x + 8x 8 5x + x x + 9 x + x + 6x + x + 9 x + x x 6x x 5x x + x 5x x + x + 0 x 9 9 x x 6 x x + 6x x x + 5x 6x + 0 x + 7x

20 6 Maths Quest 9 for Victoria Higher order expansions and Pascal s triangle Pascal s triangle, as shown in the figure below, is a special arrangement of numbers in a triangular shape. Any number is the sum of the two numbers immediately above it, with s running down the sides. The triangle was named after Blaise Pascal, a French mathematician who wrote about the triangle in 65. However, earlier mathematicians knew about the magic of this triangle. Chu Shih-Chieh, a Chinese mathematician, included an illustration of the triangle in a book in 0. There are many patterns to observe with Pascal s triangle. Let s look at one of these. 6 EXCEL Spreadsheet Expanding (x + a) n a Expand (x + ) b Which line of Pascal s triangle links to your answer for part a? Describe the pattern you have observed. a Expand (x + ). This means expand (x + )(x + )(x + ). Hint: First expand (x + ) then multiply this expansion by (x + ). b Which line of Pascal s triangle links to your answer for part a? Use the pattern you can observe to show that (x + ) x + x + 6x + x +. Use Pascal s triangle to expand each of the following. a (x + ) 5 b (x + ) 6 c (x + ) 7 d (x + ) 8 e (x + ) 9 f (x + ) 0 (You may need to copy the diagram above and add more lines to Pascal s triangle.) Let s look at what happens when we have a minus sign in the brackets. 5 Expand (x ), (x ) and (x ) by multiplying terms. 6 Describe what effect the minus sign has on the expansions. 7 Use Pascal s triangle and your observations from question 6 to expand each of the following. a (x ) 5 b (x ) 6 c (x ) 7 d (x ) 8 e (x ) 9 f (x ) 0 Extension Can you work out how to use Pascal s triangle to expand each of the following? Clearly explain how you are able to do this. a (x + ), (x + ),... (x + ) 0 b (x + y), (x + y),...(x + y) 0 c (x + ), (x + ),... (x + ) 0 d (x y), (x y),... (x y) 0

21 Chapter 5 Expanding 6 Simplifying algebraic fractions addition and subtraction In chapter we spent some time simplifying algebraic fractions. This section is effectively a continuation of that topic with more complicated fractions, requiring use of the expansion techniques we have learned in this chapter. Pronumerals in the numerator only This type of problem is very similar to those in chapter. WORKED Example Simplify the following. y + y a b y + y 7 a Write the fractions. a Find the lowest common denominator (LCD). The lowest common multiple (LCM) of and 5 is 0. 5( y + ) ( y ) ( y + ) + ( y ) Express as a single fraction. - 0 Simplify the numerator by expanding brackets and collecting like terms. b Write the fractions. b Find the lowest common denominator (LCD). The lowest common multiple (LCM) of 7 and is. 5y y y ( y + ) 7( y ) ( y + ) 7( y ) Express as a single fraction. - Simplify the numerator by expanding brackets and collecting like terms. y + y + 5 y + y 7 y + 6 7y y + 7

22 6 Maths Quest 9 for Victoria Pronumerals in the denominator In the case where there are pronumerals in the denominator, the common denominator is taken to be the product of each of the denominators. Then, proceed as in previous cases. Simplify the following. a WORKED Example 7 7 a + + a b b 5 b + a Write the fractions. a Find the LCD. The LCM of (a + ) and (a ) is (a + )(a ). 7( a ) 7( a + ) ( a + ) ( a ) + ( a + ) ( a ) 7( a ) + 7( a + ) Express as a single fraction. -- ( a + ) ( a ) Simplify the numerator by expanding brackets and collecting like terms. 7 7 a + + a 7 b Write the fractions. b --- b 5 7a 7 + 7a ( a + ) ( a ) a + 7 ( a + ) ( a ) --- b + Find the LCD. The LCM of (b 5) and (b + ) is (b 5)(b + ). 7b ( + ) b ( 5) ( b 5) ( b + ) ( b 5) ( b + ) 7b ( + ) b ( 5) Express as a single fraction ( b 5) ( b + ) Simplify the numerator by expanding brackets and collecting like terms. 8b + 7 6b ( b 5) ( b + ) b ( b 5) ( b ) It is customary to leave the denominator as brackets, without expanding them. Pairs of brackets in the denominator Although it is possible to have two different pairs of brackets in the denominator of each fraction, in this section we will consider the case where one of each pair is an identical bracket. The common denominator will consist of each bracket that appears in the question; the repeated bracket needs to appear only once.

23 Chapter 5 Expanding 65 WORKED Example Simplify. - ( x + ) ( x + ) + ( - x ) ( x + ) Write the fractions. Find the LCD. The LCM of (x + )(x + ) and (x )(x + ) is (x + )(x + )(x ). ( x + ) ( x + ) + ( x ) ( x + ) ( x ) ( x + ) ( x + ) ( x + ) ( x ) ( x + ) ( x + ) ( x ) ( x ) + ( x + ) Express as a single fraction ( x + ) ( x + ) ( x ) x 6 + x + Simplify the numerator by expanding ( x + ) ( x + ) ( x ) brackets and collecting like terms. 5x ( x + ) ( x + ) ( x ) remember remember. When adding or subtracting algebraic fractions, you must first find a common denominator.. Pronumerals can appear in either the numerators or the denominators.. If the pronumerals are in the denominator, the common denominator is usually the product of the individual denominators. 5F Simplifying algebraic fractions addition and subtraction WORKED Example Simplify each of the following. 0 a x + x + m + m + + b c d x x + y y + + e + 5 f g p + p x x + + h i j m + 6 m + p p k l m x + x x x n o x + x a + 6 a x + x x x x x SkillSHEET Adding and subtracting algebraic fractions 5.5 Mathcad

24 66 Maths Quest 9 for Victoria Simplify each of the following. m + m + x + x + 5 p p + a - - b + c + 7 y y + x + 5 x + a + a 6 d + e + f m m + p + p x x g - - h i y y 5 x x p + p + j k l y + y a a x x + m n o multiple choice a + a + a What is the equivalent of +? a + 5 7a + 7 a + 7 7a + 7 a + 5 A --- B C D E x + x b What is the equivalent of --- +? 8 x A + 6 x 5 5x 5 x + 7 x B --- C --- D --- E x + x + c The equivalent of is: 6 x + x + 8 x 5 x + A B --- C --- D -- E m m d The equivalent of is: m 7 m + m + m m A B C D E Simplify the following. 7 a b + -- c -- m + m x + x a + a + 9 d -- e f b + 5 b -- + c + c m m 7 g + -- h + -- i + -- p 8 p p 5 p a 6 a 6 j + -- k l q q a + + a + b + + b + m n o x x + m + + m - + p + p + 5 Simplify the following a b c m m + x x + x x d e f -- m + 5 m m + m p p WORKED Example

25 Chapter 5 Expanding 67 WORKED Example g -- h -- i a 7 a b b r r j k l x x + x + x + p + p m - - n o m + 7 m + p + p + x x + 6 Simplify the following. a b ( x + ) ( x + ) + ( x + ) ( x + ) ( x + ) ( x + ) + ( x + ) ( x + ) c d ( x + ) ( x + ) + ( 7 x + ) ( x ) ( x + ) ( x ) + ( x + ) ( x ) 5 e f ( x + 7) ( x ) ( x ) ( x ) ( x ) ( x ) + ( x ) ( x + ) 7 g h ( x 6) ( x ) ( x 6) ( x ) ( x + ) ( x ) + ( x + ) ( x ) 5 6 i j ( x + ) ( x + ) ( x + ) ( x + ) ( x + ) ( x + ) ( x + ) ( x + ) 5 k l ( x 6) ( x + ) ( x ) ( x 6) ( x + 7) ( x + ) ( x + 7) ( x ) m n ( x + 5) ( x 6) ( x + ) ( x 6) ( x + ) ( x + ) ( x + ) ( x 6) o p ( x 6) ( x + ) ( x + ) ( x ) ( x + 5) ( x ) ( x + 5) ( x ) 7 multiple choice a What is the equivalent of + --? x x 7 7x 6 x 6 7x 6 7x + 6 A --- B C D --- E x xx ( ) xx ( ) x xx ( ) b What is the equivalent of ? m m + 5 m + 5 m + 5 m + 5 m 5 A --- B --- C --- D --- E --- m( m+ 5) m( m+ 5) m( m+ 5) m( m+ 5) m( m+ 5) c The equivalent of --- is: x + x x + 5 6x + 6 A B --- C --- x + ( x + ) ( x ) ( x + ) ( x ) x 5 x D --- E --- ( x + ) ( x ) ( x + ) ( x ) d What is the equivalent of? ( x + ) ( x ) + ( x + ) ( x + ) x 8 A B C ( x + ) ( x ) ( x + ) ( x + ) ( x ) ( x + ) D x ( x + ) ( x ) ( x + ) E x ( x + ) ( x ) ( x + ) x ( x + ) ( x ) ( x + ) SkillSHEET Expanding 00 WorkSHEET 5.6 GAMEtime 5.

26 68 Maths Quest 9 for Victoria Expand and simplify (x 5). Expand and simplify (x y) + (5x + y). Expand and simplify (x + 5)(x + 9). Find the missing term if (y + K)(y ) y 6. 5 Expand (6 + q). 6 Expand and simplify (x + )(x + ) + (x + )(x + ). 7 Expand and simplify (x + 6)(x 5) (x ). x + x + 5 7x True or false? Simplify --. z + z Simplify. ( x + ) ( x ) ( x + 6) ( x ) Simplifying algebraic fractions multiplication and division This section is an extension of the work you did in chapter, Introductory algebra on the multiplication and division of algebraic fractions. Cancelling bracketed expressions in multiplication If a bracketed expression, or in fact any pronumeral, appears in the numerator of one term and the denominator of another, then they may be cancelled. WORKED Example Simplify each of the following. x + y a b c x + 5 y y ( z ) ( z ) ( y ) ( z 5) ( z + ) ( z + ) ( z ) ( z + ) ( z ) x + a Write the fractions. a x + 5 Cancel (x + ) and replace with a, since it appears in the first denominator and second numerator. Simplify by multiplying the remaining numerators and denominators. -- 5

27 Chapter 5 Expanding 69 b Write the fractions. b y y y ( y ) Cancel (y ) and replace with a, since it appears in the first numerator and second denominator. Cancel y and replace with a, since it appears in the first denominator and second numerator. Multiply the remaining numerators and -- 6 denominators and simplify the resultant fraction. -- c Write down the expression. c ( z ) ( z ) ( z 5) ( z + ) ( z + ) ( z ) ( z + ) ( z ) ( z ) Cancel (z ) and replace with a, since it -- ( z 5) ( z ) ( z + ) appears in the first numerator and second denominator. Cancel (z + ) and replace with a, since it appears in the first denominator and second numerator. Simplify the resultant fraction. ( z ) ( z ) ( z 5) ( z + ) As before, it is customary to leave bracketed expressions intact without expanding any further. Cancelling bracketed expressions in division When dividing fractions, we first turn the second fraction upside down and change the division sign to a multiplication sign (times and tip). We then proceed as for multiplication. WORKED Example Simplify the following. a b c x y x y 5 y ( z + ) ( z ) y ( z ) z z a Write the fractions. a x x Convert to a multiplication problem by x turning the second fraction upside down. x Cancel (x ) and replace with a Simplify the resultant fraction. -- Continued over page

28 70 Maths Quest 9 for Victoria b Write the fractions. b y y 5y y y y Convert to a multiplication problem by y 5y turning the second fraction upside down. Cancel (y ) and replace with a Cancel y and replace with a Simplify the resultant fraction c Write the fractions. c ( z + ) ( z ) ( z ) z z ( z + ) ( z ) Convert to a multiplication problem by ( z ) ( z ) ( z ) turning the second fraction upside down. ( z + ) Cancel (z ) and replace with a Cancel (z ) and replace with a. Simplify the resultant fraction. z z + So, division of fractions is the same as multiplication with an additional first step. remember remember. If a bracketed expression appears in the numerator of one term and the denominator of another, they can be cancelled when multiplying.. When dividing fractions, take the second fraction, turn it upside down and change the division sign to a multiplication sign (times and tip). 5G Simplifying algebraic fractions multiplication and division SkillSHEET WORKED 5.7 Example Simplify the following. a x + b y x + y c d b + 5 e 9 b m m m m + f p p p p 6 p 6 p

29 Chapter 5 Expanding 7 y 7 y g h i y a a y 7 a a p 5( p + ) p p ( x + ) ( x 7) x + j k l x + ( b ) ( b + ) b x 7 ( b + 5) ( b ) b x ( x + ) ( x + ) x + ( x 6) ( x ) WORKED Example Simplify the following. x + x + m m a b - - c x x x SkillSHEET 5.8 d e f a + a m m - + m 6 m - 6 p + p p p s 6 s g h i ( s + ) ( a + ) ( a ) a + s + a + a + 8m m --- ( m + ) ( m 5) m - 5 ( a + ) ( a ) a + j k l ( a 5) ( a + ) a mm ( ) m ( m + ) ( m + 5) m - + p p( p ) p + 7 ( - p + ) ( p + 7) Simplify the following. Mathcad a c ( y ) ( y + ) y y + yy ( + ) y m m( m ) m 7 ( m + ) ( m 7) m - + b d ( x + 6) ( x ) 6x x x( x+ ) x + 6 b + 7 ( b 6) ( b + 7) b b 0( b + ) b Multiplying and dividing algebraic fractions e d 8 5d( d 6) 5d ( d 7) ( d 8) d 7 f s ( s ) ( s ) -- s - 0 s( s+ 6) ss ( + 6) multiple choice m + 8 a Which is the equivalent of -? m - + A B C (m + ) D -- E x b The equivalent of is: x 9 A B 9 C ( x ) D E 9 9 m 8 m 8 c What is the equivalent of - -? m ( m 8) A --- B m C m D --- E m m ( a + ) ( a ) a + d Which is the equivalent of? a + a + A a B C a D ( a + ) ( a ) --- ( a + ) E a + a a a + 9 x m

30 7 Maths Quest 9 for Victoria Applications There are many problems in algebra where the expansion of brackets is a key component in finding a solution. A most important skill is to be able to convert a word, or real-life problem into an algebra sentence or expression. WORKED Example 5 A rectangular swimming pool measures 0 m by 0 m. A path around the edge of the pool is x m wide on each side. a Find the area of the pool. b Find an expression for the area of the pool plus the path. c Find an expression for the area of the path. d If the path is.5 m wide, calculate the area of the path. 0 m 0 m a Construct a drawing of the pool. a x b x 0 m x 0 m x Calculate the area of the pool. Area m Write expressions for the total length b Length 0 + x + x and width. 0 + x Width 0 + x + x Find an expression for the area using the formula: Area length width. c Find an expression for the area of the path by subtracting the area of the pool from the total area. d Substitute.5 for x in the expression found for the area of the path. 0 + x Area length width (0 + x)(0 + x) 0(0) + 0(x) + x(0) + x(x) x + 0x + x ( x + x ) m c Area of path total area area of pool x + x 600 (00x + x ) m d When x.5, Area of path 00(.5) + (.5) 59 m The algebraic expression found in part c of worked example 5 allows us to calculate the area of the path for any given width, x.

31 Suppose that the page of a typical textbook is cm high by 6 cm wide. The page has margins at the top and bottom of x cm and on the left and right of y cm. a Write an expression for the height of the page that is inside the margins. b Write an expression for the width of the page that is inside the margins. c Write an expression for the area of the page that is inside the margins. Chapter 5 Expanding 7 d Show that if the margins on the left and right are doubled, then the area available to be printed is reduced by (8y xy) cm. a WORKED Example 6 Construct a drawing, showing the key dimensions. a 6 cm x ANTHONY GRAY SANDRA KENMAN y y cm The real height ( cm) is effectively reduced by x cm at the top and x cm at the bottom. b Similarly the width (6 cm) is reduced by y on the left and y on the right. c The effective area of the page is the width times the height. d If the left and right margins are doubled, they become y and y respectively. Determine the new expression for the width of the page. Determine the new expression for the area. Determine the difference in area by subtracting this from the first result for the effective area. b x Height x x x Width 6 y y 6 y c Area ( x)(6 y) (6) + ( y) x(6) x( y) 8 8y x + xy d Width 6 y y 6 y Area ( x)(6 y) (6) + ( y) x(6) x( y) 8 96y x + 8xy Difference in area 8 8y x + xy (8 96y x + 8xy) 8 8y x + xy y + x 8xy 8y xy So the amount by which the area is reduced is (8y xy) cm.

32 7 Maths Quest 9 for Victoria remember remember. When working on worded problems try to convert English sentences into algebraic ones.. Drawing a diagram is an excellent aid in problem solving. 5H Applications Answer the following for each shape. i Find an expression for the perimeter. ii Find the perimeter when x 5. iii Find an expression for the area and simplify by expanding if necessary. iv Find the area when x 5. a b c d x x x + x x e x + f 5x + g x + 5 x x 6x WORKED ORKED Example 5 x + A rectangular swimming pool measures 50 m by 5 m. A path around the edge of the pool is x m wide on each side. a Find the area of the pool. b Find an expression for the area of the pool plus the path. c Find an expression for the area of the path. d If the path is. m wide, calculate the area of the path. x

33 Chapter 5 Expanding 75 WORKED Example 6 The page of a book is 0 cm high by 5 cm wide. The page has margins at the top and bottom of the page of x cm, and on the left and right of the page of y cm. a Write an expression for the height of the page that is inside the margins. b Write an expression for the width of the page that is inside the margins. c Write an expression for the area of the page that is inside the margins. d Show that if the margins on the left and right are doubled, then the area available to be printed on is reduced by (0y xy) cm. A rectangular book cover is 8 cm long and 5 cm wide. a b c d Find the area of the book cover. i If the length of the book cover is increased by v cm, write an expression for its new length. ii If the width of the book cover is increased by v cm, write an expression for its new width. iii Write an expression for the new area of the book cover and expand. iv Find the area of the book cover if v cm. i If the length of the book cover is decreased by d cm, write an expression for its new length. ii If the width of the book cover is decreased by d cm, write an expression for its new width. iii Write an expression for the new area of the rectangle and expand. iv Find the area of the book cover if d cm. i If the length of the book cover is made x times as long, write an expression for its new length. ii If the width of the book cover is increased by x cm, write an expression for its new width. iii Write an expression for the new area of the book cover and expand. iv Find the area of the book cover if x 5 cm. 5 A square has dimensions of 5x metres. a Write an expression for its perimeter. b Write an expression for its area. c i If its length is decreased by m, write an expression for its new length. ii If its width is decreased by m, write an expression for its new width. iii Write an expression for its new area and expand. iv Find its area when x 6 m. 6 A rectangular sign has a length of x cm x and a width of x cm. a Write an expression for its perimeter. b Write an expression for its area. c i If its length is increased by y cm, find an expression for its new length. ii If its width is decreased by y cm, find an expression for its new width. iii Write an expression for its new area and expand. iv Find its area when x cm and y cm. x

34 76 Maths Quest 9 for Victoria WorkSHEET 5. 7 A square has a side length of x cm. a Write an expression for its perimeter. b Write an expression for its area. c i If its side length is increased by y cm, write an expression for its new side length. ii Write an expression for its new perimeter and expand. iii Find the perimeter when x 5 cm and y 9 cm. iv Write an expression for its new area and expand. d Find the area when x. cm and y.6 cm. 8 A swimming pool with length p + metres and width p metres is surrounded by a path of width p metres. Find the following in expanded form. a An expression for the perimeter of the pool. b An expression for the area of the pool. c An expression for the length of the pool and path. d An expression for the width of the pool and path. e An expression for the perimeter of the pool and path. f An expression for the area of the pool and path. g An expression for the area of the path. h The area of the path when p m. p + p p Billboard costs At the beginning of this chapter we were looking at the increased cost of a billboard if the length and height are increased by metres and metres respectively. Does the increase in cost depend on the initial size of the billboard? Let the length of the billboard be l and the height be h. (Both l and h are in metres.) Write an expression for the area of the billboard. What is the cost of this billboard at a rate of $50 per square metre? If the length of the billboard is increased by metres, write an expression for the new length. If the height of the billboard is increased by metres, write an expression for the new height. 5 Use your answers to parts and to find an expression for the new area. 6 What is the new cost of the billboard? 7 Subtract the original cost of the billboard to find the increase in the cost. Is this cost constant or does it depend on the original length and height?

35 Chapter 5 Expanding 77 summary Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows. Expansion, in algebra, means to multiply everything the brackets by what is outside. After expanding an expression you should always look to. The expansion of 6(x + ) ( x) is. When multiplying two expressions within brackets each term in the first bracket by each term in the second bracket. 5 The expansion of ( + x)( x) is an example of a pattern called. 6 The expansion of (7 x)(7 + x) is. This follows the rule (a + b)(a b). 7 The identical bracket expansions or rules are: (a + b)(a + b) a + ab + b (a b)(a b) a ab + b. 8 When adding algebraic fractions, the first step is to find the. 9 The common denominator used when subtracting -- x x is. 0 Division of fractions is done by turning the problem into a problem. The fraction is turned when dividing fractions. Brackets can be left in the of the answer when working with algebraic fractions. When working on, you are trying to convert English sentences into algebraic ones. Drawing a diagram is an excellent aid in solving. WORD LIST denominator multiply common denominator upside down inside problem collect like terms word problems 9 x 0x + 6 perfect squares difference of two squares second multiplication x(x ) a b

36 78 Maths Quest 9 for Victoria CHAPTER review 5A Expand these expressions. a 5(x + ) b (y + 5) c x( x) d m(m + ) Expand and simplify by collecting like terms. 5A a (x ) + 9 b (5m ) c m(m ) + m 5 d 7p (p + ) Expand and simplify the following expressions. 5B a (a + b) + (a + b) b (x + y) + (x y) c m(n + 6) m(n + ) d x( x) (x ) Expand and simplify these expressions. 5C a (x + )(x + 5) b (m )(m + ) c (m )(m 5) d (a + b)(a b) 5 Expand and simplify these expressions. 5D a (x + )(x ) b (9 m)(9 + m) c (x + y)(x y) d ( a)( + a) 6 Expand and simplify these expressions. 5D a (x + 5) b (m ) c (x + ) d ( y) 7 Expand and simplify these expressions. 5E a (x + )(x + ) + (x + )(x + ) b (m + 7)(m ) + (m + ) c (x + 6)(x + ) (x + )(x ) d (b 7) (b )(b ) 8 Simplify the following. 5F x + x + m + m a + 6 a + y + 5 y a + b c d Simplify the following. 5F 5 a b - m + m m + m c d x x + y + y e f ( x + ) ( x + ) ( x + ) ( x + ) ( x ) ( x + ) ( x + ) ( x + ) 0 Simplify the following. 5G x + a b c p + x + p p a + 7 p + a a + 7 a d ( b 6) ( b + ) e f ( b + 5) ( b ) b + b m m + p ( p ) ( p + ) ( m + ) ( m + ) 9 0 p A rectangular table top has a length of x cm and a width of x cm. 5H a Write an expression for its perimeter. b Write an expression for its area. c i If its side length is increased by y cm, write an expression for its new side length. test ii Write an expression for its new perimeter and expand. yourself iii Find the perimeter when x 90 cm and y 0 cm. iv Write an expression for its new area and expand. 5 v Find the area when x 90 cm and y 0 cm. CHAPTER