Factorising FACTORISING.

Transcription

1 Ftorising FACTORISING

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3 Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will lern how ifferent kins of expressions re ftorise in their own wy. Answer these questions, efore working through the hpter. I use to think: Wht ftors 0 n 1 hve in ommon? Wht ftors o 0 n 1 hve in ommon? ( x- )( x+ ) x - 4. Complete the following: x - 16 ( x-?)( x +?) How mny ftors is qurti trinomil ftorise into? Answer these questions, fter working through the hpter. But now I think: Wht ftors 0 n 1 hve in ommon? Wht ftors o 0 n 1 hve in ommon? ( x- )( x+ ) x - 4. Complete the following: x - 16 ( x-?)( x +?) How mny ftors is qurti polynomil ftorise into? Wht o I know now tht I in t know efore? 100% Ftorising Mthletis 100% 3P Lerning 1

4 Ftorising Bsis Common Ftors Whole numers tht multiply together to give prtiulr numer re lle the ftors of tht numer. For exmple, the numer 6 n e expresse s the prout of two whole numers in two wys: 6 6# 1 n 6 # 3. So the ftors of 6 re: 1,, 3, 6. A ommon ftor of two numers is numer tht is ftor of oth numers. Here is n exmple of fining the highest ommon ftor. Wht is the highest ommon ftor (HCF) of 1 n 18? To get the highest ommon ftor, first get the ftors of eh numer. Then pik the highest one. 1 1 # 1 1 6# # # 1 4# # 3 Ftors of 1 re 1,, 3, 4, 6, 1. Ftors of 18 re 1,, 3, 4, 9, 18. ` The highest ommon ftor (HCF) of 1 n 8 is 6. Ftors our in lger too. In lger the expression mens #. So n re ftors of. Here re some exmples of ommon ftors in lger. Wht is the ommon ftor of eh of these: n. The ommon ftor is, sine it is ftor of oth terms. xy n xz. The ommon ftor is x. A ommon ftor of n is. But lso ours in oth terms, so is lso ommon ftor. The highest ommon ftor of n is. Here is nother exmple. Wht is the highest ommon ftor of xy 4 n xz? 4 xy x # x y # xz x z 4 Rememer tht x x # x. x is the highest power of x whih ours in oth 4 xy n xz. 4 ` The highest ommon ftor of xy n xz is x. In term suh s 10xy, the numer 10 is lle the oeffiient. The oeffiient is the numer in front. Here is n exmple. Wht is the highest ommon ftor of 0xyz 4 n 1x 3 z? The highest ommon ftor of the oeffiients, 0 n 1, is The highest ommon ftor of xyzn xzis xz 3. Therefore putting these together, the HCF is 4xz 3. x 3 is the highest power of x whih ours in oth terms. z is ommon to oth. 100% Ftorising Mthletis 100% 3P Lerning

5 Ftorising Bsis Ftorising Ftorising is the opposite of expning. The gol of expning is to remove rkets from expressions, like this exmple: ( + ) + The gol of ftorising is to put expressions into rkets, like this exmple: + ( + ) Using Common Ftors Common ftors re si to e ftorise out of expressions. In the expression +, the vrile is ommon in oth terms. So is ommon ftor of oth terms. This mens n e ftorise out of the expression s ove. Here re some exmples: Ftorise these expressions: 5xy + 5y 5yx ( + 1) xy-4xy xy( x-y) 5 y is ommon in oth terms xy is ommon in oth terms In ove, oth 5 n y re ommon ftors on their own, ut 5 y is the highest ommon ftor (HCF). In, xy is the highest ommon ftor. To fin the HCF, fin the highest ommon numeril ftor of the oeffiients n then the lowest ommon power of eh vrile. Ftorise these expressions: ( + 3 ) 1xy-4xy xy( 3y -x) is the HCF 4x y 3 is the HCF 3 is the HCF of the oeffiients 3 is the lowest power of 3 is the lowest power of 4 is the HCF of the oeffiients x is the lowest power of x y 3 is the lowest power of y Sometimes there will e more thn two terms. Then the HCF must e foun for ll the terms! Ftorise this expression: ( ) 4 is the HCF of the oeffiients is the lowest power of 3 is the lowest power of 100% Ftorising Mthletis 100% 3P Lerning 3

6 Ftorising Bsis Brkets s Common Ftors Sometimes rket is ommon ftor. If this hppens then the whole rket is ftorise out. Look t this exmple: Ftorise the following: xx ( + 3) + 5yx ( + 3) ( x + 3) is ommon in oth terms ( x+ 3)( x+ 5y) Ftorising using Grouping n Brkets The ove exmple my not hve een groupe into rkets, n my hve een in expne form Ftorise the following (y grouping in pirs): x + 6x+ 5xy + 15y Mny expressions, like in the ove line, oul hve 4 (or more) terms without ommon ftor ross ll terms x -9xy- 4x+ 6y is ommon 4 is ommon 3 x is ommon - is ommon ( + ) + 4 ( + ) 3 x(x-3 y) -(x-3) y ( + ) is ommon rket ( x - 3y ) is ommon rket ( + )( + 4 ) ( x-3y)( 3x - ) Sometimes the expression nees to e reorere efore it n e groupe to fin ommon rket. Here is n exmple: Ftorise the following into two rkets: Step 1: Reorer. Step : Fin ommon ftors. Step 3: Ftorise out the ommon rket ( - ) + 3( - ) ( - )( + 3) Chek this y expning the rkets to see if you get the originl expression % Ftorising Mthletis 100% 3P Lerning

7 Ftorising Questions Bsis 1. Fin the HCF of these terms: 6x n x 5 8x n 6x y 1 n ef, 10 ef n 16ef. Ftorise the HCF out of the following: 6x+ y 4-8pq 10q - - 3pq+ 1pq 4 4 e 10xy-40xy 4 4 f g 6pq+ 3p -15pq h 7mn - 14mn+ 1mn % Ftorising Mthletis 100% 3P Lerning 5

8 Ftorising Questions Bsis 3. Ftorise the ommon rket out of the following: 3x ( x- 1) + 5y( x- 1) 4xxy ( 3 - y ) + ( xy 3 - y ) 4. Ftorise these expressions into two rkets using grouping: xy + y+ x x + 3x + 4xy + 6y mn 3 m 3 m + + n % Ftorising Mthletis 100% 3P Lerning

9 Ftorising Questions Bsis e 3 8xy + 4x -4y - xy f p + 3p q+ 4p+ 6pq g 4mn + 3mn -4m- 3mn 4 h 4 3 y+ xy + y + xy i 18pq+ 1pq + 6pq+ 4pq 3 4 j % Ftorising Mthletis 100% 3P Lerning 7

10 Ftorising Knowing More Expning rkets proues wht is lle qurti trinomil. Look t this exmple: ( x+ )( x+ 3) x + 5x+ 6 The expression on the right is qurti trinomil euse it hs three terms n the highest power is. In generl, qurti trinomil looks like this: x x + + Leing Coeffiient (in front of x ) Coeffiient of x onstnt (no x ) Rememer, ftorising is the opposite of expning. So ftorising qurti trinomil mens hnging it k to rkets. Ftorising these trinomils is one ifferently epening on the vlue of (the leing oeffiient). Ftorising Qurti Trinomils if 1 If 1then the qurti trinomil looks like this: x x + + To ftorise these expressions, two numers re neee suh tht: 1. Their prout is (the onstnt). Their sum is (the oeffiient of x) n These two numers re written in the rkets. Here re some exmples: Ftorise these qurti trinomils into two rkets: x + 5x+ 6 Step 1: Fin two numers with prout of 6 (the onstnt) n sum of 5 (the oeffiient of x). # 3 6 n Prout Sum So the numers re + n +3. Step : Ftorise y writing these two numers in rkets with x. x + 5x+ 6 ( x+ )( x+ 3) x -5x-6 Step 1: Fin two numers with prout of -6 (the onstnt) n sum of -5 (the oeffiient of x). 1 # (- 6) - 6 n 1 + (- 6) -5 Prout Sum So the numers re +1 n -6. Step : Ftorise y writing these two numers in rkets with x. x -5x- 6 ( x+ 1)( x- 6) The orer of these numers is not importnt 8 100% Ftorising Mthletis 100% 3P Lerning

11 Ftorising Questions Knowing More 1. Fin vlues for m n n if they hve these sums n prouts: m+ n 3 n mn m+ n 7 n mn 1 m+ n 0 n mn -4 m+ n - 3 n mn e m+ n 3 n mn -10 f m+ n - 3 n mn -18. Answer these questions out the trinomil x + 9x+ 0: Write own the vlue of the leing oeffiient (), the oeffiient of x () n the onstnt (). Fin two numers whose prout is () onstnt n whose sum is (). Ftorise the trinomil into two rkets. 3. Answer these questions out the trinomil x -4x- 1: Write own the vlue of the leing oeffiient (), the oeffiient of x () n the onstnt (). Fin two numers whose prout is () onstnt n whose sum is (). Ftorise the trinomil into two rkets. 100% Ftorising Mthletis 100% 3P Lerning 9

12 Ftorising Questions Knowing More 4. Ftorise these trinomils into rkets: x + 4x + 3 x + x- 6 x - 6x + 5 x + x- 4 e x - 6x - 16 f x - 14 x+ 40 g x - 5x - 14 h x - 1 x % Ftorising Mthletis 100% 3P Lerning

13 Ftorising Knowing More Ftorising Qurti Trinomils if! 1 Ftorising x + x + when is not 1 is slightly more omplite. This will e shown with n exmple. Ftorise 1x + x- 4 into two rkets Step 1: Fin the vlue of # # # - - Step : Fin the two numer whih hve prout of # (-48) n sum of (). 8 # (- 6) - 48 n 8 + (- 6) So the two numers re 8 n -6 Step 3: Rewrite the originl (oeffiient of x) in terms of the two numers in Step. 1x + x -4 1x + 8x-6x -4 The 'x' is reple y 8x- 6x Step 4: Use grouping to ftorise. 4 x(3x+ ) - (3x + ) ( 3x+ )( 4x- ) Here is nother exmple: Ftorise 1y + 10y - 1 into two rkets Step 1: Fin the vlue of # # # - - Step : Fin the two numer whih hve prout of # (-4) n sum of (-10). # (- 1) - 4 n + (- 1) -10 So the two numers re n -1 Step 3: Rewrite the originl (oeffiient of y) in terms of the two numers in Step. 4y - 10y -1 4y + y-1y -1 Step 4: Use grouping to ftorise. 4y - 1y+ y -1 1 y(y- 1) + (y -1) ( y- 1)( 1y+ 1) 100% Ftorising Mthletis 100% 3P Lerning 11

14 Ftorising Questions Knowing More 5. Look t 16x + 1x+. Ientify, n. Fin the vlue of #. Fin two numers whih hve prout equl to # n sum equl to. Ftorise 16x + 1x+ into two rkets. 6. Look t 6x -x-. Ientify, n. Fin the vlue of #. Fin two numers whih hve prout equl to # n sum equl to. Ftorise 16x -x- into two rkets % Ftorising Mthletis 100% 3P Lerning

15 Ftorising Questions Knowing More 7. Ftorise these qurti trinomils: x + 5x+ 1m + 5m p + 3p + 3 5t + 37t + 14 e 8k - k + 5 f % Ftorising Mthletis 100% 3P Lerning 13

16 Ftorising Knowing More Differene of Squres Rememer when expning rkets tht iffer in sign only ^x+ yh^x- yh x -y. This mens tht if qurti expression is in the form x - y then it n e ftorise esily into ^x+ yh^x-yh. Herer re some exmples. Ftorise the following into two rkets using the ifferene of squres: x m - 5n x -4 ( m) -( 5n) ( x+ 4)( x- 4) ( m- 5n)( m+ 5n) Don t get onfuse: x this wy. + y n t e ftorise into two rkets. Only ifferene of squres n e ftorise Perfet Squres Rememer, for perfet squres ( x+ y) ^x+ yh^x+ yh x + xy+ y. This mens tht if qurti expression is in the form x + xy+ y then it n esily e ftorise into ( x+ y). Here re some exmples. Ftorise the following: m + 6m+ 9 x - 4x+ 4 m + ( m)( 3) + 3 x + ()( x - ) + (-) ( m + 3) ( x -) p - 6pq+ q ( 3) + ( 3)( ) + ( ) ( 3p) + ( 3p)( - q) + (-q) ( 3+ ) ( 3p-q) Whenever you ftorise nything, you n lwys hek y expning the rkets to see if you get the originl expression % Ftorising Mthletis 100% 3P Lerning

17 Ftorising Questions Knowing More 8. Ftorise these expressions: x + 14x + 49 y - 9 4y + 8y+ 4 16q - 36p e 81p - 11q f 64m + 3mn+ 4n g 49t - 56tu+ 16u h 9xy+ 1pqxy+ 4p q 100% Ftorising Mthletis 100% 3P Lerning 15

18 Ftorising Using Our Knowlege Using more thn one Metho to Ftorise The methos lerne up until nee to e omine sometimes to ftorise ertin expressions. Ftorise 4y -1y-16 Metho 1 Metho 4y 1y 16 4 y -1 y -16 Step 1: Ftor out ommon ftor Step 1: Ftorise the trinomil 4( y -3y- 4) 4y 16y 4y # 4-64( # ) 4yy ( - 4) + 4( y -4) () ( y- 4)( 4y+ 4) Step : Ftorise the qurti trinomil Step : Ftorise the ommon ftor out from the seon rket 4( y- 4)( y+ 1) 4( y- 4)( y+ 1) Sometimes the ifferene of squres metho nees to e use more thn one, or omine with ftorising out ommon ftor. Here re some exmples. Ftorise the following: m - n 4 4 3x 108x 3 - ( m ) -^n h ^m + n h^m - m h ^m + n h^m+ nh^m-nh Rememer: only ifferene of squres n e ftorise this wy 3xx ^ -36h 3xx ^ + 6h^x - 6h Differene of squres Common ftor If there re four terms or more, then grouping my hve to e use with other methos. Here is n exmple. Ftorise p + 4p- q + 4q p + 4p- q + 4q p - q + 4p+ 4q ^p+ qh^p- qh+ 4( p+ q) Differene of squres Common ftor ^p+ qh6 ^p- qh+ 4@ ^p+ qh^p- q+ 4h % Ftorising Mthletis 100% 3P Lerning

19 Ftorising Using Our Knowlege Ftorising Trinomils with more thn one Vrile. Wht hppens if qurti trinomil like x -xy- 6y or x + 7xy- 15y nees to e ftorise? If the first n lst terms re oth squre, then the sme is proess for ftorising trinomils is followe. Ftorise x -xy-6y Step 1: Fin two numers with prout of - 6y (oeffiient of en term) n sum of - 1y (oeffiient of mile term). - 3y# y 6y n - 3y+ y -1y Prout Sum So the numers re - 3y n + y. Step : Ftorise y writing these two numers in rkets with x n y. x -xy- 6y ^x- 3yh^x+ yh Ftorise x + 7xy- 15y Step 1: Step : Fin the prout of the first oeffiient n the lst oeffiient. # Fin the two numers with prout of the ove numer n sum of the mile oeffiient. 10 # (- 3) - 30 n 10+ ^- 3h 7 So the two numers re 10 n -3 Step 3: Rewrite the originl mile oeffiient in terms of the two numers in Step. x + 10xy-3xy- 15y x - 3xy+ 10xy-15y Step 4: Use grouping to ftorise. x^x- 3yh+ 5y^x-3yh ^x- 3yh^x+ 5yh Sometimes, of ourse, this metho of ftorising will e omine with nother. Ftorise 3pq+ pq-4pq pq+ pq- 4pq pq^3p + pq- 4q h pq^3p - 3pq+ pq-4q h pq6 3p^p- qh+ 4q^p-qh@ - 3# 4-1 ( # ) () pq^p- qh^3p+ 4qh 100% Ftorising Mthletis 100% 3P Lerning 17

20 Ftorising Questions Using Our Knowlege 1. Ftorise the following: 10m + 5m x + 6 x+ 40 x - y 8x -xy-y 8 8 e 3 8t -6t -10t f 3x + 5xy-y % Ftorising Mthletis 100% 3P Lerning

21 Ftorising Questions Using Our Knowlege g 10x x h i -4x - xy+ 1y j ( y-3) -( y- 5) k m 4n 4n m l - 1t + 7u -1t-18u 100% Ftorising Mthletis 100% 3P Lerning 19

22 Ftorising Thinking More Algeri Frtions Numeril frtions n e simplifie y fining ommon ftors in the numertor n enomintor. For exmple 1 18 # 6 3# 6 Algeri frtions hve lgeri expressions in the numertor n the enomintor. They n lso e simplifie y nelling ommon ftors. To o this, oth the numertor n enomintor nee to e ftorise. Here re some exmples: Simplify these lgeri frtions: 3 x - 8 x + 4 y -y-15 3y ( y- 5) + ( y - 5) ( x - 4) ( x + ) Common ftor ( y+ 3)( y-5) 3y ( y- 5) + ( y - 5) ( x+ )( x-) ( x + ) Differene of squres ( y+ 3)( y-5) ( y - 5)( 3y + ) ( x+ ) ( x-) ( x + ) Cnel ommon rkets ( y+ 3)( y-5) ( y - 5)( 3y + ) x - y + 3 3y + To simplify ny lgeri frtion, ftorise the numertor n enomintor n remove the ommon rkets n ftors. Simplify these lgeri frtions s muh s possile: 3 x -3x - 8x + 1 x + x-6 ( m-n) -( m+ 3n) m + 10mn+ 5n x ( x-3) -4 ( x - 3) ( x- 3)( x+ ) 6( m-n) -( m+ 6( m- n) + ( m+ ( m+ 5n) ( x-3)( x -4) ( x- 3)( x+ ) (-m- 5n)( 3m+ n) ( m+ 5n)( m+ 5n) (x- 3)( x+ )( x - ) (x- 3)( x+ ) -( m+ 5 n)( 3m+ n) ( m+ 5 n)( m+ 5n) x - -( 3m+ n) m+ 5n -3m-n m+ 5n 0 100% Ftorising Mthletis 100% 3P Lerning

23 Ftorising Questions Thinking More 1. Simplify these frtions s muh s possile: m + 14m+ 4 m + 3 9x - 4y 9x + 10xy+ 4y 3 4t - t + 4t - 1 4t % Ftorising Mthletis 100% 3P Lerning 1

24 Ftorising Thinking More Multiplying n Diviing Frtions Algeri frtions re multiplie n ivie the sme wy s numeril frtions. Before multiplying or iviing, ftorise eh frtion s muh s possile. Then rkets n e nelle to mke lultions esier. Here is n exmple multiplying two lgeri frtions Multiply these two lgeri frtions 4-1 # Ftorise the numertor n enomintor in oth frtions. ( + )( -) ( + 3)( -4) # ( - 4)( + 5) ( + )( -3) Cnel the ommon rkets. ( + )( -) ( + 3)( -4) # ( - 4)( + 5) ( + )( -3) ( - )( + 3) ( + 5)( - 3) Here is n exmple iviing two lgeri frtions Fin the following quotient 8x -6x-5 ' x + 7x+ 3 x 4x x+ 9 Ftorise the numertor n enomintor in oth frtions. ( x+ 1)( 4x-5) 4x - 5 ' ( x+ 3)( x+ 1) ( x + ) 3 Flip the seon frtion to rete prout of frtions. Cnel the ommon rkets. ( x+ 1)( 4x-5) ( x + 3) # ( x+ 3)( x+ 1) 4x - 5 ( x+ 1)( 4x-5) ( x + 3) # ( x+ 3)( x+ 1) 4x - 5 x % Ftorising Mthletis 100% 3P Lerning

25 Ftorising Thinking More Aing n Sutrting Frtions When ing n sutrting lgeri frtions, ommon enomintor must e foun. Simplify the following (y omining the frtions): 4-3 x x 10 3y y y 8-3 x x 4 4# 8 x x# x 10y 3y y y # y 10y y y# y y x 10y+ 3y+ 1 y 5 x 13y + 1 y Here re two exmples where the enomintors hve ifferent letters. Simplify the following (y omining the frtions): x x 1# + 1# # # ^5x- 1h # x + # 7 7 # x x # 7 + ^5x- 1hx+ 14 7x + + Rememer tht s the orer of multiplying oesn't mtter. 5x - x+ 14 7x Here re two more exmples where ifferent numers or letters our in the enomintor. Simplify the following (y omining the frtions): x 3x m 5n 3 # x + 5# 8 8x # 3 3x # 8 5# 5n + 3# 7m 7m# 5n 5n# 7m x 5n + 1m 35mn 35mn 49 4x 5n+ 1m 35mn 100% Ftorising Mthletis 100% 3P Lerning 3

26 Ftorising Thinking More The next two exmples involve enomintors with two or more terms. Simplify the following (y omining the frtions): 3 4 x+ + 5x+ 1 3# ^5x + 1h 4# ^x + h + ^x + h# ^5x + 1h ^5x+ 1h# ^x+ h 35 ^ x+ 1h+ 4^x+ h ^x+ h^5x+ 1h 15x+ 3+ 4x+ 8 ^x+ h^5x+ 1h 19x + 11 ^x + h^5x + 1h 3 p - 5p 3 + p -3p-10 p + Hint: Ftorise the enomintor 3 p - 5p + 3 ( p + )( p - 5) p + 3 p - 5p 3 ( p - 5) + ( p + )( p - 5) ( p + )( p - 5) Ftorise enomintor Common enomintor 3 p - 5p + 3 ( p - 5) ( p+ )( p-5) 3 p - 5p + 6p - 15 ( p+ )( p-5) p ( p- 5) + 3 ( p - 5) ( p+ )( p-5) Grouping in pirs ( p- 5)( p + 3) ( p+ )( p-5) (p- 5) ( p + 3) ( p+ )(p-5) Cnel ommon rkets p + 3 p % Ftorising Mthletis 100% 3P Lerning

27 Ftorising Questions Thinking More. Simplify the following prouts n quotients: y - 3x - 3xy # 3x- 3y y - y m + 13m-7 m 14m # - m - 49 m - 1 p + 5 ' p - p + 10p+ 5 p - 4p x 6+ 11x+ 4x ' - x 6-11x+ 4x 100% Ftorising Mthletis 100% 3P Lerning 5

28 Ftorising Questions Thinking More e # f 4x- 3y 8x + 10xy-3y ' x+ y 4x - 9y 6 100% Ftorising Mthletis 100% 3P Lerning

29 Ftorising Questions Thinking More 3. Simplify these expressions: x x+ 1 6x 4 100% Ftorising Mthletis 100% 3P Lerning 7

30 Ftorising Questions Thinking More 4x + + x - 4 x + 1 x + y 3y y y % Ftorising Mthletis 100% 3P Lerning

31 Ftorising Questions Thinking More e 3 3 x+ - x+ 1 f 1 ( y+ 3)( y+ 4) - ( y+ 3)( y-4) 100% Ftorising Mthletis 100% 3P Lerning 9

32 Ftorising Questions Thinking More g p p 3p + 5p+ - p + 1 h m m m m+ 30 m % Ftorising Mthletis 100% 3P Lerning

33 Ftorising Questions Thinking More i x+ 1 + x+ + x - 1 j 4- ' % Ftorising Mthletis 100% 3P Lerning 31

34 Ftorising Answers Bsis: Knowing More: 1. x n ef + 3 n + 4. (3 x+ y) - ( ) q(4pq - 5) 4 3 pq( - q + 4 p ) e f - n + - n n n + 3 e 10 xy( y - 4 x ). 1, 9, 0 f ( + 3 ) + 5 n + 4 g p(pq + p -5 q ) ( x+ 5)( x+ 4) h mn (1- mn+ 3 m) 3. 1, - 4, 1 3. (x- 1)(3x + 5) y + 3 n -7 3 ( xy- y )(4x+ ) ( x+ 3)( x-7) 4. ( x+ 1)( y+ ) 4. ( x+ 3)( x+ 1) ( - )( + 4) ( x+ 3)( x-) ( x+ 3)( y+ x) ( x-5)( x-1) ( mn+ 1)( 3+ m) ( x+ 6)( x-4) e ( 4x- y)( y + x) e ( x- 8)( x+ ) f ( p+ 3pq)( p+ ) f ( x-4)( x-10) g ( 4m+ 3mn )( n -1) g ( x+ )( x-7) h yxy ( + 1)( y + 1) h ( x-8)( x-4) i pq( 3p + q)( 3+ p) j -( - )( + 1) % Ftorising Mthletis 100% 3P Lerning

35 Ftorising Answers Knowing More: Using Our Knowlege: 5. 16, 1, 1. 5 ( m- 1)( m+ 1) 3 -( x- 10)( x+ 4) + 8 n ( x + y )( x + y )( x+ y)( x-y) ( 8x+ )( x+ 1) ( 4x+ y)( x-y) 6. 6, - 1, n -4 ( 3x- )( x+ 1) e f g h i t( 7t- 5)( t + 1) ( 3x- y)( x+ y) 10xx ( - 10)( x + 10) ( + )( + )( - ) -(( x+ y)( 4x- 6y)) 7. ( x+ )( x+ 1) j ( y - 8) ( 3m- 1)( 4m+ 3) k ( m+ n)( m-n- ) ( 7p+ 1)( p+ 3) l - 3 ( t+ 3u)( t- 3u+ ) 8. ( 5t+ )( t+ 7) ( 4k-1)( k-5) ( 3-5)( -3) ( x + 7) ( y+ 3)( y-3) ( y + ) 1. Thinking More: ( m + 4) ( 3x) - ( y) 9x + 10xy+ 4y t + 1 ( - + 4) ( - ) e ( 4q- 6p)( 4q+ 6p) ( 9p- 11q)( 9p+ 11q). x y f ( 8m+ n) m g h ( 7t- 4u) ( 3xy + pq) ( p - ) ( p + 5) 1 ( 4x-3)( x-) # ( 4x + 3) 1 100% Ftorising Mthletis 100% 3P Lerning 33

36 Ftorising Answers Thinking More:. e f ( + ) ( - ) ( 4x-3y)( x-3y) ( x+ y)( 4x-y) e f g h i j x - 3x 6 1x - 9x ( x + 1) 3y ( y + 1) 3( x - 1) ( x+ )( x+ 1) ( y - 8) ( y+ 3)( y+ 4)( y - 4) - p ( 3p+ )( p+ 1) 3 m - 1 ( m + 3) 3x + 4x-1 ( x+ 1)( x+ )( x - 1) ( - ) % Ftorising Mthletis 100% 3P Lerning

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