MEP Y8 Practice Book A. In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below: ( ) = +

Transcription

1 8 Algebra: Brackets 8.1 Epansion of Single Brackets In this section we consider how to epand (multiply out) brackets to give two or more terms, as shown below: = First we revise negative numbers and order of operations. Eample 1 Evaluate: (a) (b) (c) ( 6) ( 5 ) (d) (e) (f) 6( 8 15) (g) 3 ( 5) (h) (a) = 4 ( ) 3 1 (b) 7 + ( 4) = 7 4 = 11 (c) ( 6) ( 5) = 30 = ( ) (d) = 18 (e) 4 ( 8 + 3) = 4 11 = 44 = ( ) (f) = 4 17

2 8.1 MEP Y8 Practice Book A (g) 3 ( 5) = = 8 (h) ( ) = 3 1 = 1 1 = 1 When a bracket is epanded, every term inside the bracket must be multiplied by the number outside the bracket. Remember to think about whether each number is positive or negative! Eample Epand 3( + 6) using a table. From the table, Eample 3 Epand 4 ( 7) = = = 4 8 Remember that every term inside the bracket must be multiplied by the number outside the bracket. Eample 4 Epand ( 8 ). 18

3 ( 8 ) = 8 = 8 Eample 5 ( ) Epand 3 4. ( 3) ( 4 ) = ( 3) 4 ( 3) = 1 ( 6 ) = Eercises 1. Calculate: (a) (b) 6 14 (c) 6 5 (d) 6 9 (g) 8 7 (j) (e) ( ) ( ) 11 4 (f) 6 4 (h) 88 ( 4) (i) 6( 8 10) (k) 711 ( 4) (l) ( 4) ( 6 17). Copy and complete the following tables, and write down each of the epansions: (a) (b) = 5( 7) = 4 + (c) 3 (d) = 5( + 5) =

4 8.1 MEP Y8 Practice Book A 3. Epand: (a) (b) 3( 4) (d) 7( 3 4) (c) (f) 8( 3 9) (e) (g) ( ) ( 4) (h) ( 3) 8 (i) (j) 9( + 8) =. 4. Jordan writes Eplain why his epansion is not correct. 5. Copy and complete the following tables and write down each of the epansions: (a) (b) y ( ) = = y 6. Copy the following epansions, filling in the missing terms: = +? (b) ( )( ) =? + (a) (c) =? (d) 6 ( 7) = 6? =? (f) ( )( + ) =? (e) 3 y 3 7. Epand: (a) ( ) (c) 6 + (e) + y 7 (b) (d) 4 ( 3 5) (f) ( 4y 3) (h) 5 ( y 1) (g) + 3y 8. Write down epressions for the area of each of these rectangles, and then epand the brackets: 1 (a) (b)

5 (c) (d) (e) (f) Write down an epression for the area of this triangle, that: (a) contains brackets, (b) does not contain brackets Write down an epression for the volume of this cuboid, that: (a) contains brackets, (b) does not contain brackets Linear Equations Epanding a bracket will usually be the first step when solving an equation like Eample 1 Solve 5( 3) = 35 4( + 3) = 0 5( 3) = 35 Epanding brackets gives: 5 15 = 35 Adding 15 to both sides gives: 5 = 50 Dividing by 5 gives: =

6 8. MEP Y8 Practice Book A Eample Solve 6( + 7) = 50 6( + 7) = 50 Epanding brackets gives: = 50 Subtracting 4 from both sides gives: 6 = 8 Dividing by 6 gives: = 8 6 = Eample 3 Gilda thinks of a number and adds 7 to it. She then multiplies her answer by 4 and gets 64. (a) Write down an equation that can be used to calculate the number with which Gilda started. (b) Solve your equation to give the number. (a) Start with. Add 7 to give + 7 Multiply by 4 to give 4( + 7) This epression equals 64, so the equation is 4( + 7) = 64 (b) 4( + 7) = 64 Epanding brackets gives; = 64 Subtracting 8 from both sides gives: 4 = 36 Dividing by 4 gives: = 36 4 = 9 13

7 Eercises 1. Solve these equations: = (b) 5( 8) = 40 (a) = (d) 7( + 4) = 4 (c) = (f) 3( 4) = 11 (e) = (h) 10( + 7) = 8 (g) Solve these equations: (a) = (b) 33 ( + 6) = 7 = (d) 8( 1) = 4 (c) A rectangle has sides of length 3 m and + 4 m. Find the value of, if the area of the rectangle is 18 m. 3 m ( + 4) m 4. Feti chooses a number, adds 7, multiplies the result by 5 and gets the answer 55. (a) If is the number Feti first chose, write down an equation that can be used to determine the number. (b) Solve the equation to determine the value of. 5. The following flow chart is used to form an equation: (a) Write down the equation. (b) Solve the equation to find the value of. 6. Solve the following equations: = = (a) 47 0 (b) = = (c) (d) = = (e) (f)

8 8. MEP Y8 Practice Book A 7. Alice thinks of a number, subtracts it from 11 and then multiplies her answer by 5 to get 45. What was the number that Alice started with? 8. Solve the following equations: 9. = ( ) (b) 3( + 4) = 11 (a) = ( + ) (d) 47 ( ) = 5( + ) (c) m (a) ( + 4) m Write down an epression for the area of the triangle. (b) What is if the area is 15 m? 8.3 Common Factors As well as being able to remove brackets by epanding epressions, it is also important to be able to write epressions so that they include brackets; this is called factoring or factorisation. Eample 1 Factorise First write each term as a product of factors: = { = + 3 [Note that is the only factor common to both terms and is placed outside the brackets.] Now you can check your answer by epanding it. 134

9 Eample Factorise 18n n + 4 = 3 3 n = 6( 3n + 4) Note that both and 3 are factors of both terms, and so 3 = 6 is placed outside the brackets. Eample 3 Factorise = + 3 = + 3 Note that both and are factors of both terms, and so = is placed outside the brackets. Eample 4 Factorise = = Note that because 5 and are factors of both terms, a 1 must be introduced in the bracket when the 5 is placed outside the brackets. You can check the calculation 'backwards': = =

10 8.3 MEP Y8 Practice Book A Eample 5 Factorise 3y + 1 y 3y + 1 y = 3 y y y = 3y y+ 4 Note that 3, and y are factors of both terms, and so 3 placed outside the brackets. y = 3y is Eercises 1. Factorise: (a) + 4 (b) (c) (d) 5 5 (e) 3 1 (f) (g) 9 1 (h) (i) Factorise: (a) 3 + (b) (c) 6 3 (d) 6 4 (e) (f) Denise states that (a) (b) = Is her statement true? Describe how it could be improved. 4. For each statement below, decide if it has been fully factorised and if not, complete the factorisation: (a) + = +1 (c) 5 30 = 5 30 (e) 6 18 = 3 6 (b) = (d) 8 3 = 4 8 (f) 15 6 =

11 5. Eplain why the following factorisation is incorrect: = Factorise: (a) y+ z (b) yz+3yz (c) 4pq 8qr (d) 5yz+ 0uy (e) 5y 4py (f) 7y+ 1z 7. Factorise: (a) y + y (b) 3 y + 6y (c) 5 y 35y (d) y+ 4 y 3 (e) yz + y z (f) y z 6 3 (g) y + y (h) y + y (a) Epand ( + y + z). (b) Factorise 5 + y + 4z. 9. Factorise: (a) 3 + 9y + 18z (b) y (c) 6 3y + 1z (d) 5z y (e) y 1y (f) 4 + 6z + 15y 10. Factorise: 3 (a) 4y+ 1 y+ (b) 6 y 4 y y (c) 3 y 4y + y (d) 5 7 y y z 137

12 8.4 Epansion of Two Brackets When two brackets are multiplied together, for eample, ( + ) + 3 every term in the first bracket must be multiplied by every term in the second bracket. Eample 1 Use a table to determine ( + 3) ( + ) + 3 ( + ) The multiplication table is formed using the two brackets. The contents of the table give the epansion. ( + ) ( + 3 ) = or + 3 = = Eample Use a table to determine ( 6) + ( + ) ( 6) So, ( 6) ( + ) = or + = =

13 An alternative method for epanding two brackets is shown in the net eample. Eample 3 Determine ( + ) 7 ( + ) ( 7 ) = ( 7) + 7 = or 7 = = 5 14 Note how each term in the first bracket multiplies the whole of the second bracket. Eercises 1. Copy and complete the following tables and write down each of the epansions: (a) (b) ( + 4) ( + 5) ( + 4) 7 (c) 4 (d) 5 1. Epand: ( 1) ( + 4) ( ) 5 ( + ) ( + ) (a) (b) 5 ( ) ( ) (c) 5 1 (d) ( ) ( ) (e) + 3 (f)

14 8.4 MEP Y8 Practice Book A 3. Epand: ( + ) ( ) (a) 1 1 (b) + ( + ) ( + ) (c) 5 5 (d) 7 7 How are the answers to this question different from the others you have done? 4. Eplain what is wrong with this statement: ( + 5) = Epand: (a) ( + 1) (b) ( 1) (c) ( + 3) (d) ( 5) 6. (a) Copy and complete this table: 6 1 (b) What is the epansion of ( + 1) ( + 6)? 7. Epand: ( + ) (b) ( 3 + 1) ( 4+ 1) (a) ( + ) (d) ( 4 1) ( 5 + 1) (c) 1 3 (f) ( 4 3) (e) Write out the following epansions, filling in the missing terms: ( + ) = + + = + + (a) + 7 6? 4 (b) + 6? 36 ( ) = + + ( + ) = (c) 5? 10 (d) 1 1? ( + ) = + + = + (e) + 3 1? 7 3 (f) 7? Eplain what is wrong with this statement: ( ) =

15 10. Write out the following epansions, filling in the missing terms: ( ) = + (a) +? 1 ( ) = (b) + 4? 4 ( + ) = + + (c) + 3? 9? ( + ) = (d)? 5? ( + ) = + + (e) +?? 4 4 ( + ) = + + (f) +?? The following eample shows how to determine ( + 1) ( + ) 1 = = ( + ) = = = Use the same method to determine: (a) ( + 1) 4, (b) ( + 1) 5. Compare your answers with Pascal's Triangle and describe any connections that you see. 141