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WorkSHEET 7.1 Factorising and expanding Name: _________________________ 1 Expand and simplify the following:

(a) (4x – 1)(2 – 3x) (b) (x + 4)(1 + x) − (x + 3)( x – 5)

2

2

(a) (4 1)(2 3 )8 12 2 312 11 2

x xx x xx x

− −

= − − +

= − + −

( ) ( )

19715245

153544)5)(3()1)(4()b(

22

22

+=

++−++=

−+−−+++=

−+−++

xxxxx

xxxxxxxxxx

2 Expand and simplify the following: 2)37()a( +x

22 )4(2)3()b( xx −−+

( )( )

942499212149

3737)37()a(

2

2

2

++=

+++=

++=

+

xxxxx

xxx

( )( ) ( )( )

( )( )

2 2

2 2

2 2

2 2

2

(b) ( 3) 2(4 )3 3 2 4 4

3 3 9 2 16 4 4

6 9 2 16 8

6 9 32 16 222 23

x xx x x x

x x x x x x

x x x x

x x x xx x

+ − −

= + + − − −

= + + + − − − +

= + + − − +

= + + − + −

= − + −

3 Expand and simplify the following: (a) (x – 5)(x + 5) (b) (2a – 1)(2a + 1)

(a) (x – 5)(x + 5)

= x2 – 25 (b) (2a – 1)(2a + 1)

= 4a2 – 1

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4 Expand and simplify the following: (a) 3(m – 3)2 (b) 2(w – 4)(w + 4)

(a) 3(m – 3)2

= 3(m2 – 6m + 9) = 3m2 – 18m + 27

(b) 2(w – 4)(w + 4)

= 2(w2 – 16) = 2w2 – 32

5 Factorise: (a) x2 – 49i2 (b) 5x2 – 125

(a) 22 49ix −

( )( )ixix 77 −+= (b) 1255 2 −x

( )( )( )555

255 2

−+=

−=

xxx

6 Factorise: (a) 3x2 – 6x + xy – 2y (b) 14a2b – 7ab2 + 2a − b

(a) 3x2 – 6x + xy – 2y

= 3x(x – 2) + y(x – 2) = (x – 2)(3x + y)

(b) 14a2b – 7ab2 + 2a − b

= 7ab(2a – b) + (2a – b) = (2a – b)(7ab + 1)

7 Factorise: (a) x2 – 7x – 18 (b) b2 – 9b – 36

(a) 1872 −− xx

= (x – 9)(x + 2) (b) b2 – 9b – 36

= (b – 12)(b + 3)

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8 Factorise: (a) 3x2 – 14x – 5 (b) 5g2 + 8g – 21

(a) 3x2 – 14x – 5

= 3x2 – 15x + x – 5 = 3x(x – 5) + (x – 5) = (x – 5)(3x + 1)

(b) 5g2 + 8g – 21

= 5g2 + 15g – 7g – 21 = 5g(g + 3) – 7(g + 3) = (g + 3)(5g – 7)

9 Factorise: (a) x2 – 6x + 9 (b) 4x2 + 12x + 9

(a) x2 – 6x + 9

= (x – 3)2 (b) 4x2 + 12x + 9

= 4x2 + 6x + 6x + 9 = 2x(2x + 3) + 3(2x + 3) = (2x + 3)2

10 Factorise the following by looking for a common factor first: (a) 2x2 – 14x + 24 (b) 6x2 + 27x – 15

(a) 2x2 – 14x + 24

= 2(x2 – 7x + 12) = 2(x – 3)(x – 4)

(b) 6x2 + 27x – 15

= 3(2x2 + 9x – 5) = 3(2x2 + 10x – x – 5) = 3(2x(x + 5) – (x + 5)) = 3(x + 5)(2x – 1)