«A first introduction to symmetrical algebraic expressions»

Transcription

1 «A first introdution to smmetril lgeri epressions» Bkground informtion: In mthemtis ompetitions t ll level, from seondr shool to Interntionl Mthemtil Olmpid(IMO), prolems relted to smmetril properties re ommon ple This is first ttempt to introdue this topi to lower seondr students who re keen to tke prt in n of these ompetitions So, let s enjo Wht is smmetril epression? The degree of n epression is onept similr to tht of polnomil For these vriles,,,, these re degree epressions:,, + + These re degree epressions: 4, +, ( + ),,, These re degree epressions:, +, ( + ),,, + + This is degee smmetril epression: + + Wht it mens is tht if is ehnged with either or, the epression remins the sme + + is degree smmetril epression is degree smmetril epression So, + + is degree smmetril epression of vrile, Give tht,, re onstnts;,, re vriles,,, re similr terms;,, re similr terms;,, re similr terms +, degree smmetril epression of, vriles + +, degree smmetril epression of, vriles + + +, degree smmetril epression of, vriles Golden propert of smmetril epressions: For two smmetril epressions, the opertions of sum, differene, produt, nd division, ield smmetril epression Pge of 9 O level Mth Note KLAng June 0

2 Let s egins with few worked emples: Emple : Epnd is degree smmetril epression of vrile, the produt is still smmetril epression is degree smmetril epression of vrile, Degree vriles, 4 4,, 4 4 Let, where,, re onstnts Equting term, When, When,, 7, 4 4 Solve the two equtions,, We n use Binomil Theorem to show tht, Emple : Epnd is degree smmetril epression of vrile,, Degree vriles, while for lerner to list these vriles orretl,, It m tke Let re onstnts, where,, Equting term, When,, 0, Pge of 9 O level Mth Note KLAng June 0

3 When,,, 6 6 Emple : Simplif is degree smmetril epression of vrile,, is degree smmetril epression of vrile,, is lso degree smmetril epression of vrile,, is degree smmetril epression of vrile,, When, 0 is ftor of the epression Degree vriles, is degree smmetril epression of vrile,,, Let, where, re onstnts When,, 0, () 9 When,,, () Solving () nd () simultneousl, we get 0, Pge of 9 O level Mth Note KLAng June 0

4 Emple 4: Simplif is degree 4 smmetril epression of vrile,, is lso degree 4 smmetril epression of vrile,, is degree smmetril epression of vrile,, is degree smmetril epression of vrile,, When 0, 0 the epression So, smmetr, is ftor of is degree smmetril epression of vrile,, Let where is onstnt When,,, 6 Emple : Ftorise When, 0 So, is ftor of this epression When 0, 0 epression, is lso ftor of the epression Let p q So, is ftor of this epression Sine this is smmetril where p, q re onstnts Pge 4 of 9 O level Mth Note KLAng June 0

5 When, When,, p q p q (), p q Solving () nd (4), we get p, q p q (4) Emple 6: Ftorise After some trils nd errors, we n find tht is ftor of this epression Let p q When, 0, 0 When,, 0 where p, q re onstnts, p p, q q It remins hllenge in ftorising n lgeri epression Guessing orret ftor of n epression is still ver muh dependent on the pst eperiene Wht is rottionl smmetril epression? The epression is not smmetril If is ehnged with, the epression eomes tht is not the sme s the originl epression However, if is hnged with, is ehnged with nd is ehnged with, then, the epression remins the sme is lled rottionl smmetril epression Some of these rottionl smmetril epressions:,, ww u Golden propert of rottionl smmetril epressions: For two rottionl smmetril epressions, the opertions of sum, differene, produt, nd division, ield nother rottionl smmetril epression Pge of 9 O level Mth Note KLAng June 0

6 Pge 6 of 9 O level Mth Note KLAng June 0 For degree nd ove vrile of,,, there is different tretment etween nonrottionl nd rottionl smmetr For non-rottionl smmetr, we write, As for rottionl smmetr, it is modified, d Eh group is rottionll the sme Of ourse, rottionl smmetr group is lso sugroup in the smmetr group Let s stud few emples: Emple 7: Epnd is degree rottionl smmetril epression of vrile,, Let s r q p, where p, q, r, s re onstnts Sine does not hve term, 0 p When,, 0, r q 0 r q () When,, 0, r q (6) Solving () nd (6), we get r, q When,,, s r q 0 0 s Emple 8: Simplif is degree rottionl smmetril epression of vrile,,

7 is degree rottionl smmetril epression of vrile,, When, 0 rottionl smmetr, When is ftor of the epression B re lso ftors of the epression, we get degree 0 epression, whih is onstnt Let k, where k onstnt When,, 0, 6 k k Emple 9: Ftorise When, 0 So, is ftor of this epression Sine this is rottionl smmetril epression, nd re lso ftor of the epression Let f,, where f, rottionl smmetril epression f, p q, is degree,, the degree rottionl smmetr is the sme s the smmetr ones When,, 0 When,,, p q p q (7), 6p q Solving (7) nd (8), we get p, q 0 6p q (8) Pge 7 of 9 O level Mth Note KLAng June 0

8 Pge 8 of 9 O level Mth Note KLAng June 0 The net few emples will minl del with frtionl epressions Emple 0: Simplif The given frtion is rottionl smmetr After some rerrngement, we get When, 0, therefore,, re ftors of the numertor Let p When,, 0, p p Emple : Given tht,,, prove tht When eh given frtion is hnged from to to, the others re hnged from to to The given frtions re rottionl smmetries

9 Pge 9 of 9 O level Mth Note KLAng June 0 B rottionl smmetr, we get In onlusion: In this rtile, I hve introdued the si properties of smmetril epressions in reltion to epnsion, simplifition, nd ftorition, through the emples Needless to s, one of the most importne s well s hllenge is in ftorition Guess nd hek, with the demonstrted pprohes, re served s leds in eh ftoristion As smmetr hs prtiulrl importnt ple in ompetition mthemtis, I will most likel e revisiting this topi gin soon