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WS2 Mixed factorisation 1 Factorise –25x(a – 3n) – 15xy(a – 3n).

I like Finney’s way better … watch my video if you have forgotten my easy way to do this *** The text book makes it look harder than it actually is! HCF = −5x(a – 3n) Divide each term by HCF. −25x(a – 3n) ÷ −5x(a – 3n) = 5 −15xy(a – 3n) ÷ −5x(a – 3n) = 3y Complete the factorisation. –25x(a – 3n) – 15xy(a – 3n) = −5x(a – 3n)(5 + 3y)

2 Factorise 8x2 – 12xy –2px + 3py.

I like Finney’s way better … watch my video if you have forgotten my easy way to do this *** The text book makes it look harder than it actually is! There are no common factors in all 4 terms. Group terms in pairs. The common factor of 8x2 and −12xy is 4x. 8x2 – 12xy = 4x(2x – 3y) The common factor of –2px and 3py is −p. –2px + 3py = −p(2x – 3y) Complete the factorisation.

)4)(32()32()32(432128 2

pxyxyxpyxxpypxxyx

−−=

−−−=+−−

3 Factorise (x + 1)2 – y2.

Factorise, using the rule for difference of two squares. (x + 1)2 – y2 = (x + 1 + y)(x + 1 – y)

4 Factorise 36x2 – 25y2.

Factorise, using the rule for difference of two squares.

)56)(56()5()6(

253622

22

yxyxyx

yx

−+=

−=

−

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5 Factorise a2 + 24a – 25.

Find all the factor pairs of −25 and their sum. Factors: 1, −25 : sum = −24 5, −5 : sum = 0 −1, 25 : sum = 24* Find the pair (marked *) whose sum equals the coefficient of the middle term (24). Factorise, using the rule for quadratic trinomials. a2 + 24a – 25 = (a – 1)(a + 25)

6 Factorise 6x2 – 5x – 6.

Find all the factor pairs of –36 and their sum.

0 sum :6 6,5 sum :9 4,

*5 sum :9 4,9 sum :21 3,

9 sum : 21 3,61 sum :81 2,

61 sum :81 2,35 sum : 63 1,

35sum :63 1,:Factors

=−

=−

−=−

=−

−=−

−=−

=−

=−

−=−

The factor pairs 4 and –9 multiply to give –36 and add to give –5 (marked *).

( ) ( )( )( )3223

2332326946656 22

−+=

+−+=

−−+=−−

xxxxx

xxxxx

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7 Complete the square for each of the following expressions: (a) xx 82 + (b) mm 62 −

(a) x2 + 2× (4)x

= x2 + 2× (4)x +16−16

= x + 4( )2 −16 (b) m2 − 2× (3)m

= m2 − 2× (3)x + 9− 9

= m−3( )2 − 9

8 Factorise the following by completing the square:

158)a( 2 ++ xx

2012)b( 2 +− xx

I like Finney’s way better … My way is done in the previous question! … watch my video if you have forgotten my easy way to do this *** The text book makes it look harder than it actually is!

)3)(5()14)(14(

1)4(1516)168(

158218

218

158)a(

2

2

222

2

++=

−+++=

−+=

+−++=

+⎟⎠

⎞⎜⎝

⎛ ×−⎟⎠

⎞⎜⎝

⎛ ×++=

++

xxxx

xxx

xx

xx

)10)(2()46)(46(

16)6(2036)3612(

20122112

2112

2012)b(

2

2

222

2

−−=

−−+−=

−−=

+−+−=

+⎟⎠

⎞⎜⎝

⎛ −×−⎟⎠

⎞⎜⎝

⎛ −×+−=

+−

xxxx

xxx

xx

xx

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9 Factorise the following by completing the square:

52)a( 2 −− xx

36)b( 2 −+ xx

*** I like “Finney’s way” better J ***

)61)(61(

6)1(51)12(

52212

212

52)a(

2

2

222

2

−−+−=

−−=

−−+−=

−⎟⎠

⎞⎜⎝

⎛ −×−⎟⎠

⎞⎜⎝

⎛ −×+−=

−−

xx

xxx

xx

xx

)323)(323(

)123)(123(

12)3(39)96(

36216

216

36)b(

2

2

222

2

−+++=

−+++=

−+=

−−++=

−⎟⎠

⎞⎜⎝

⎛ ×−⎟⎠

⎞⎜⎝

⎛ ×++=

−+

xx

xx

xxx

xx

xx

10 Factorise each of the following: (a) 20x2 – 4x (b) x2 – 13 (c) 3x2 + 14x – 5

(a) 20x2 – 4x

= 4x(5x – 1) (b) x2 – 13

= (x + 13 )(x – 13 ) (c) 3x2 + 14x – 5

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