Binomials and Pascal s Triangle
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1 Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00
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3 Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst powr). (ii) Th highst offiint. Th offiints r,, 8,,,,. Thrfor, th highst offiint is. (iii) Th onstnt trm. Th onstnt trm is. Sin th onstnt trm is th trm without. (iv) Th lowst offiint. Th offiints r,, 8,,,,. Thrfor, th lowst offiint is. (v) Th offiint of. Th offiint of is 0. Sin thr is no trm ontining. (vi) Writ own th trm involving. (Th trm inlus sign, offiint n itslf). (vii) Th offiint of. 8. Intif th prts of th polnomil:. (i) Th gr. Th gr is. (Sin is th highst powr). (ii) Th offiint of. Th offiint of is 0. Sin thr is no trm ontining. (iii) Th onstnt trm. Th onstnt trm is. Sin th onstnt trm is th trm without. 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
4 Binomils n Psl s Tringl Bsis. Intif th prts of th polnomil: (iv) Th highst offiint.. Th offiints r,,,. Thrfor, th highst offiint is. (v) Th offiint of. Th offiint of is. (vi) Th lowst offiint. Th offiints r,,,. Thrfor, th lowst offiint is. (vii) Th offiint of. Th offiint of is.. Simplif, prssing th nswr without rkts. ^h ` ^h 0 0 o 0 mn ` ^ m n h m n o f o m 0 ^h B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
5 Binomils n Psl s Tringl Knowing or. Epn n simplif. ^ h 8 ^ h `8 8 ` f m ` 0 0 m 0 0 g ` h m 0 0 m. Epn n simplif. ^ h ^ h ^h ^ h ^ h ` m f f m 8 p 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
6 Binomils n Psl s Tringl Knowing or. Epn n simplif. ^ h^ h ^ h^ h ^ h ^h ^h ^h ^ h^ h ^ h^ h ^h ^h ^ h^ h 0 8 ^ h^ h f ^ h^ h ^ h^ h ^ h^ h 8 g ^ h^ h h ^ h^ h 8 ^ h^ h ^ h ^ h 0 i ` ` ^ h^ h ^ h ^ h 0 k ` ` l m m ` ` m m 8 8 B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
7 Binomils n Psl s Tringl Knowing or. Complt th list of offiints low. Eprssion Coffiints p p p z z ^0. hz z z z % Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
8 Binomils n Psl s Tringl Knowing or 8. Complt ths pnsions of inomil powrs irt lultion [o not us formuls]: (i) ^ h ^ h^ h (ii) ^ h ^ h^ h ^ h^ h using (i) (iii) ^ h ^ h ^ h ^ h^ h using (i) Complt ths pnsions of inomil powrs irt lultion [o not us formuls]: (i) ^ h ^ h^ h ^h ^h (ii) ^ h ^ h^ h ^ h^ h using (i) ^ h ^h ^h ^ h (iii) ^ h ^ h ^ h ^ h^ h using (i) ^ h ^h ^ h B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
9 Binomils n Psl s Tringl Using Our Knowlg. Complt Psl s Tringl low. Rmmring tht ll th numrs on th lft n right sis r ll th sm numr; wht is tht numr? Using Psl s Tringl from qustion, writ own th pnsions of th following inomil powrs. ^ h Complt th rst of this pnsion. [Hint: It s th sm s for prt pt th signs ltrnt] ^ h ^p h ^ph ^ph 0 ^ ph 0^ph ^ph p 0p 0p 0p p ^ h ^h ^h ^h ^h ^ h ^h ^h ^ h ^h ^ h ^h ^h % Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
10 Binomils n Psl s Tringl Using Our Knowlg. Writ own th pnsions using Psl s Tringl n simplif: [Hint: Rfr to Psl s Tringl from qustion ] ^ ph ^ h p p p p 0 0 ^P h P P P 0P P P ^p mh p p m p m p m p m p m pm m p mp m p m p m p m p m p m ^ h Writ own th pnsions using Psl s Tringl n simplif: [Hint: Rfr to Psl s Tringl from qustion ] ^ ph ^h ^h ^ph ^ h ^ph ^h^ph ^ph p p p 8p ^ h 8^h ^ h ^h 0 ^ h ^h 0 ^ h ^h ^h^h ^hb ^P h 8P P P 0P P P B P P 0P 80P 0P P ^p mh 8p p ^mh p ^mh p ^mh p ^mh p ^mh p^mh ^mh B p mp m p 0m p 80m p 0m p 0m p 8m ^ p mh 8^ ph ^ ph ^ mh 0 ^ ph ^ mh 0 ^ ph ^ mh ^ ph^ mh ^ mhb p mp 0m p 0m p 00m p 80 m B 8 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
11 Binomils n Psl s Tringl Thinking or. Epn n simplif using Psl s Tringl. ` ` ` 0` 0` ` m ` ` m ` m ` m m 8 8 ` 0 ` ` ` ` ` ` ` ` ` ` ` ` ` ^h ^h ` ^ h ` ^ h` ` 8 ` 0 0 ` ` ` ` ` % Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
12 Binomils n Psl s Tringl or Qustions. Intif th following prts for h of th polnomils: (i) Th gr. (i) Th gr. (ii) Th highst offiint. (ii) Th highst offiint. (iii) Th lowst offiint. (iv) Th onstnt trm. 0 (iii) Th lowst offiint. (iv) Th onstnt trm. (v) Th offiint of 8. (v) Th offiint of 8. 0 (vi) Th offiint of. (vii) Th trm involving. (viii) Th offiint of. 0 (vi) Th offiint of. 0 (vii) Th trm involving. (viii) Th offiint of. (i) Is th polnomil moni? Ys, sin th offiint of th highst powr is qul to. (i) Is th polnomil moni? No, sin th offiint of th highst powr is not qul to. () Th ling trm. () Th ling trm. B 0 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
13 Binomils n Psl s Tringl or Qustions. Simplif th following, prssing our nswr without rkts: ` 8 ^ h ` ^h ^h 8^h o o ^ h ^h f f p m ^h ^h ^h ^ h ^h ^8h ^h o ^ h ^h 0 A A A. Epn n simplif: 8 m ` 0 8 o ` 0 m 8 8 ` 0 8 m. Epn n simplif: g g g g 0 0 ` ` g`g g g g g ^gh g g 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
14 Binomils n Psl s Tringl or Qustions. Epn n simplif: s s s m s s s s s s m s 8 s ss f o ` ` p. Epn n simplif: ^ h h hh h m h h h h ; h h h h E h h 0h h m ` 0 8 `g g`g g g g g ` g g g g `g 8 g g g g 8 g g g g 0 m 0 m m 0 0 m List ll th offiints of th polnomil for powrs of p from 0 to :,,,,, 0, 0. Th offiint of p is ^h whih is. Also, th p trm offiint n onstnt r oth 0. p p p p ^ph. B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
15 Binomils n Psl s Tringl or Qustions. Epn ths irt lultion or using Psl s Tringl: ^z h ^m h ^z h^z h ^m h^m h z z z z m m m ^h m ^h z z m m ^T h ^T h^t h^t h ^T h^t T T T h ^T h^t T h T T T T T T T T T T ^ h ^ h^ h@ ^ Epn th following using Psl s Tringl: ^k h k k ^ h 0k ^ h 0k ^ h k^ h ^h k 0k 0k 80k 80k ^ h ^h ^h ^h ^h 8 08 ^g h ^gh ^gh ^ h ^gh ^ h 0^gh ^ h ^gh ^ h ^ gh^ h ^ h 0g g 80g 80g 0g g 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
16 Binomils n Psl s Tringl or Qustions 8. Epn th following using Psl s Tringl: ^ ^ h ^h ^h ^h 0^h ^h 0^h ^h ^ h^h f ^ w uh 8^wh ^ wh ^ uh ^wh ^ uh ^ wh^ uh ^uh B w 000w u 00w u 0wu 8w 000uw 800u w 80u w 8u g ^ R Ph 8^ Rh ^ Rh P ^ Rh P ^ Rh P ^ Rh P ^ Rh P ^ RhP P B 080R 80PR 880P R 00P R 800P R 0P R P R P. Writ own th first 0 lins of Psl s Tringl B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
17 Binomils n Psl s Tringl or Qustions 0. Us Psl s Tringl to pn th following inomil powrs. ^ h Th powr of tlls us to us th offiints,, 0, 0,, from Psl s Tringl, so th pnsion oms: 0 0 ^ h Th powr of tlls us to us th offiints,,,, from Psl s Tringl, so th pnsion oms: ^h ^h ^h ^h 8 ^A Bh Th powr of tlls us to us th offiints,,, from Psl s Tringl, so th pnsion oms: ^Ah ^Ah ^ Bh ^Ah^ Bh ^Bh A 8A B AB B Evr son trm is ngtiv us of th () minus sign in th mil. ^ h ` ^ h ^ h 0 ^ h ^ h ^ h ^h ^ h ^ h `^ h ^ h ^ h 0 ^ h ^ h 0 ^ h ^ h ^ h^ h ^ h ` f ^Q Ph `Q 8Q ^Ph 8Q ^Ph Q ^Ph 0Q ^Ph Q ^Ph 8Q ^Ph 8Q^Ph ^Ph 8 8 Q Q P Q P 8Q P 0Q P Q P Q P 0QP P 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
18 Binomils n Psl s Tringl or Qustions. Epn n simplif using Psl s Tringl: ` ` ` ` ` 8 `w u ^wh ^wh u 0 w u 0 w u w u u ` ^ h ` ^ h ` ^ h` ` w 0uw 0u w 80 u w 80 u w u ` `` 0 0 ` ` ` ` ` ` ` ` ` 8 ` 8` ` ` ` ` ` ` ` B B 8 8 B 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
19 Binomils n Psl s Tringl or Qustions. Epn n simplif using Psl s Tringl: ` ^h ^h 0 ` ^ h ` ^ h ` ^ h ` ^ h` ` 0 ` ^h ^h 8 ` ^ h ` ^ h ` ^ h ` ^ h ` 8 8 ^ h ` ^ h ` ^ h` ` ` ` ` ` ` ` m ` 0 0 `^ h ^ h ` ^ h ` ^ h ` ^ h` ` % Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
20 Binomils n Psl s Tringl or Qustions. Whih st of numrs r th offiints for th polnomil? Th offiint of th trm is, th offiint of th is, th offiint of th is, th offiint of th trm is 0 n th onstnt is. Thrfor th nswr is,,, 0,. Wht is th gr of th polnomil:? Th trm with th highst powr is. Hn, gr. Thrfor th nswr is. Wht is th offiint of in th polnomil: 8? Sin th os not ppr in th prssion it mns tht th offiint is 0. Thrfor th nswr is 0. Wht is th pn form of th prssion? mn ^ kh mn ^ khm n^mh k mn mk Thrfor th nswr is mn mk. Wht is th pn form of th prssion? ` ` ` Thrfor th nswr is 8. Wht is th pn form of th prssion? ^ p qh^ h ^p qh^ h p p ^ h q q ^h p p q q Thrfor th nswr is p p q q B 8 00% Binomils n Psl s Tringl SERIES TOPIC NUBER thltis 00% P Lrning
21 Binomils n Psl s Tringl or Qustions. Whih on rprsnts Psl Tringl of numrs? K hrtristis of Psl' s Tringl inlu smmtr of numrs ithr si. An, most importntl, th numr low is lws th sum of th two numrs irtl ov. Thrfor th nswr is 0. Whih is th orrt pnsion of ^A Bh? From Psl's Tringl th offiints r,,,,. Sin it is ^A Bh (i w hv B trm) thn ltrnt trms r ngtiv. Thrfor th nswr is A A B A B AB B. Wht prt orrspons to th inomil pnsion: m? Sin th son trm is ngtiv i m th offints ltrnt. Hn,,, 0, 0,,. (Eliminting istrtors C n D). Appling th Binomil Epnsion th trm is ris to th fifth powr, thn fourth n so on. Distrtor B os not follow this pttrn n, hn, is inorrt. Thrfor th nswr is 0 0 ` ` m ` m ` m ` m m. Wht is th pn form of th prssion: `? ` ^ h` ` Thrfor th nswr is 00% Binomils n Psl s Tringl thltis 00% P Lrning SERIES B TOPIC NUBER
22 Binomils n Psl s Tringl
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