SUMS OF nth POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.


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1 SUMS OF th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece of the th powers of the roots of a give quadratic equatio is of i t e r e s t to the algorithmatist both for the umber patters to be see ad for the implied coectio betwee quadratic equatios ad the factorability ad primeess of u m b e r s. I this paper, a elemetary approach which poits out relatioships which might otherwise be m i s s e d is used to derive geeral expressios for the subject equatio ad to show how Lucas ad Fiboacci umbers a r i s e as special c a s e s. Sums of like powers of roots of a give equatio have bee studied i detail. For a few examples, i this Quarterly S. L. Basi [l~j has used a developmet of Warig's formula [ ] for sums of like powers of roots to write a geeratig fuctio for Lucas u mbers ad to write the kth Lucas umber as a sum of biomial coefficiets. R. G. Buschma [J has used a liear combiatio of like powers of roots of a quadratic to study the differece equatio u, = au + bu, arrivig at a result which e x p r e s s e s u i t e r m s of a ad b. aul F. Byrd ["j, [ ] h a s studied the coefficiets i expasios of a  alytic fuctios i polyomials associated with Fiboacci u m b e r s. A umber of the results i the foregoig refereces appear as special cases of those of the preset paper. A. THE SUMS OF THE th OWERS Give the primitive equatio f(x) = x  px q =, with roots r, ad r?, the first problem cosidered h e r e is to write the equatio with roots r ] ad r?. formula, i \, *. * f(x) = x  (r + r ) x  ( r x r ) =, Solvig /the primitive equatio for r, ad r by the quadratic
2 9 SUMS OF th OWERS OF ROOTS OF A GIVEN r x = ( + J? Z + q)/, r = (p  Jp Z + q)/. Sice r ad r satisfy the equatio (x  r )(x  r ) =, x  (r + r )x + r r = x  px  q = idetically, so r, + r = p ad r r~ = q. Next, write the equatio f(x), havig roots. r, ad r Squarig roots, r J = [(p + \/p + q)/] = (p + q + p / p + q)/ r\ =[(p  V/ + q)/] = (p + q  p / p + q)/. Addig, r, + r? = p + q. The desired equatio is the f(x) = x  (p + q)x + q =. Similarly, the coefficiet of x for the equatio f(x) =, by multi plyig ad addig, is r + r = p + pq. Cotiuig i this maer, the coefficiets for x for the equatios whose roots a r e higher powers of r, ad r ca be tabulated as follows: The sequece for r, + r eds whe, for a give, the last t e r m i the sequece becomes equal to zero. The peultimate t e r m a s s u m e s the form q ' whe is evead pq ' whe is odd. Writig the umerical coefficiets from Table i the form of a a r r a y of m colums ad rows ad lettig C(m, ) deote the c o efficiet i the m t h colum ad th row leads to Table. It is obvious that.the iterative operatio i Table to geerate additioal um e r i c a l coefficiets is C(m, ) = C ( m , ) + C(m, ), which suggests a liear combiatio of biomial coefficiets. expasio, By direct (  m ) (  m  l )... (m+) _?,+lm,,m ( (ml)i " m  } " ( m  l
3 QUADRATIC EQUATION April TABLE : r * + r r l + r =,, r T r =, r l + r =, t r = r l + r =, r l + r = 8, 8 r l + r = 8 + q + pq + p q + q + p q + :pq' + p q + 9p q + q + p q + p q + pq + 8p q + p q + lp q + q., Z, (  ) , (  )(  )  r l = p q q p q ~T! T (  m)(  m  )... (  m + ) m+ m ( m  )! q TABLE : C(m, ) m L SC(m, ) L L L L L L 9 8 L 9 L 8
4 9 SUMS OF th OWERS OF ROOTS OF A GIVEN F r o m the above tables, (A. ). r + l r [a/] i= z^: )  (  i  ) s ' i i p q where [ x ] is the greatest iteger less tha or equal to x, ad ( ) is the biomial coefficiet roof: By algebra, (m)l! k+ k+ k, k+ A k+ k, k x, k t k. r, + r = (r, + r~ + r r 9 + A r l ^ ' )  ( ^r, + r, ^ )' = (r^ + r ^ ) ( r i + r )  r ^ r * " + r k " l ) = p(r + r ) + q(r + r ). 'I F r o m Table, (A. ) holds for =,,..., 8. To prove (A. ) by mathematical iductio, a s s u m e that (A ) holds whe = k ad = k . Usig the result just give ad the iductive hypothesis, [k/] k+, k+ v r, + r = p i= [(kl)/] + q i= ( k :S. ( k_i " ) i i ki i q k  l  i _ ki I I k  l  i i q Multiplyig as idicated ad usig the idex substitutio i for i i the secod s e r i e s yields k+, k+ r + l r [V] i= ( k : i )  ( k " i _ ) ' k+li i q +l)/] + i [( k :j>  ( k ; ; ) i= k+li i q
5 QUADRATIC EQUATION April Sice the r e c u r s i o formula for biomial coefficiets is m m m+ l ' l + l ; { + l ' ' combiig the s e r i e s above will yield [ ( k + i  i } _ ( k  i } ] for the coefficiet of p q. Cosiderig the s e r i e s for k eve ad for k odd leads to [(k+l)/] k+, k+ v r x + r = I i= ' ( k + l  i j _ (ki } ' k+li i p q so that (A. ) follows by mathematical iductio. A expressio equivalet to (A. ) was give by Basi i [! ] Usig formula (A. ), we ca write the desired equatio,... v f(x) = x  (r x + r ) x + () q =. Example: Give the equatio x  x + =, write the equatio whose roots a r e the fourth powers of the roots of the give equatio, without solvig the give equatio. I the give equatio, p =, q =. F r o m Table or formula (A. ), r^ + rjj = () + () () + () = 9, so f(x) = x  9x + =. As a check, by factorig x x + =, r ] = ad r_ =, givig us f(x) = (x  )(x  ) = x  9x +. Returig to Table, the summatio of the coefficiets by rows yields the s e r i e s,,,,, 8, 9,,..., the successive Lucas umbers defied by L, =, L =, ad L = L, + L.   The geeral equatio for r, + r? reduces to the umerical values of C(m, ) of Table for p = q = i the primitive equatio x  px q = c p, which becomes x  x  = with roots a = ( + /ft)/ ad ( = (  \/~)/. This yields a expressio for the th Lucas umber, sice L = a + ;
6 9 SUMS OF th OWERS OF ROOTS OF A GIVEN (A. ) L^ = + + (N)/! + ()()/! + ()()()/! [/] i= ( : i )  ( 'Y l y For example, L, = + + ()/ + ()(l)/ = 8. Also, the equatio whose roots a r e the th roots of x  x  = is x  L x + () =. (Essetially, equatio (A. ), a s w e l l a s (A. ), (B. ), (B ), appears i [ ] by Buschma. ) A alterate expressio for L is obtaied by expressig r + r i t e r m s of p ad A, where A = \/p + rq e Now, T i _LA\/? o  /. ~ l. (l) ~. (l)() , r x = L( +A )/J = (p + p A +  ^ T " A + / 'p + r, \ / o i o~ / _ = l.. (l) ,(l)() , r = [( A )/J (  A + " ^ T ~  A +^r + which, for p = q =, A = v, implies, o addig, (A. ) L = ~ (l  (l)/i + (~ l)()()/! +... ) [/] _i,, T ^;.)?  l x r Z i= B. THE DIFFERENCES OF THE th OWERS Now let us tur to the equatio whose roots a r e r ad  r where r ad r a r e the roots of the primitive equatio x px q = Ct Sice r l " r = ( p + ^ p + r q ) / / " ( p " V pl + q ) / / = ^ p + q = A ' the equatio whose roots a r e r, ad ~r is x  y p + rq x  r, r =, ad f(x) = x  A x + q =. Similarly, *\ x \ = O + / + ( )/ ]  [(  V/ + <>/ ] =
7 Q U A D R A T I C E Q U A T I O N A p r i l C o t i u i g i t h i s m a e r, a d a s s e m b l i g r e s u l t s, we h a v e T A B L E r l " r A. A (p + q ) A ( p + pq)a ( p + p q + q ) A ( p + p q + p q ) A ( p + p q + p q + q )A., . _.  ^ (  ) (  )  (p + ()p q +  i JJ ' p q +..., (  m  l ) (  m  ),., (  m )  m  l m.. + _ Li ' i L p q ) A mi The sequece for r  r? as give i Table eds, for a give, o arrivig at t e r m z e r o. Writig the umerical coefficiets from Table i the form of a a r r a y of m colums ad rows, ad lettig C (m, ) deote the coefficiet i the mth colum ad th row, TABLE : C (m, ) A m % C A (m, ) L L L L L L L 8 L L
8 9 SUMS OF th OWERS OF ROOTS OF A GIVEN It is obvious that the iterative p r o c e s s to geerate additioal umerical coefficiets r e s t s o the relatio the same as for Table C A (m? ) = C A ( m  l, ) + C A (m, ), The coefficiets geerated a r e also the coefficiets appearig i the Fiboacci polyomials defied by f + (x) = x f j x ) +f _ (x), f (x) = l, f (x) = x. (For further refereces to Fiboacci polyomials, see Byrd, [ ], p.. ) Tables ad lead to l>/] (B.l) r j  r j = A (""j" ) ^ ' V. j= which is proved similarly to (A. ) The equatio whose roots a r e r ' ad  r?, where r ad r,u r\ i \ / \ J_ a r e the roots of x  px  q =, is f(x) A = x  (r  r )x + q =. We use ( B. l ) to solve the followig: i L Give the equatio x  lox + = with usolved roots r, ad r?, write the quadratic equatio whose roots a r e r, ad  r? F r o m the give equatio, p =, q = , r ad A = /  () =, Usig ( B. l ) or Table, r l " r = ( p + q ) = ( " ) = l e Hece f (x) A = x  lx + () =. As a check, the roots of the prob lem equatio a r e r ] =, r =, so f(xl = (x  )(x + ) = x  lx  =. The summatio of t e r m s i the successive rows of Table is see to yield the sequece of Fiboacci u m b e r s, defied by F, = F? =, F = F, + F o8 This result ca be demostrated by substitutig &   p = q = i the geeral expressio (B. ) ad dividig by A = /, for the left had m e m b e r becomes the Biet form, F = (a  p")/ ST. I geeral,
9 8 QUADRATIC EQUATION April (B.) F = + () + ()()/! + ()()()/! +... j=o the sum of t e r m s i risig diagoals of a s c a l ' s triagle. Also, the quadratic equatio whose roots a r e a ad (, th powers of the roots of the quadratic equatio x  x  = with roots a = ( + / ) / ad ( = (  / ) /, is x  /" F x  =. Expressig r,  r? i t e r m s of oly p ad A ad assemblig t e r m s as we did for r, + r i (A. ), we a r r i v e at a alterate exp r e s s i o for the th Fiboacci umber, (B. ) F = ~ ( I (l)()/! + ( l)()(~)()/! +... [(l)/] R < X " ( i + l J i= REFERENCES le S. L Basi, "A Note o Warig's Formula for Sums of Like owers of Roots", 9. Fiboacci Quarterly :, April, 9, pp.. L. E. Dickso, F i r s t Course i the Theory of Equatios, Joh Wiley, 9. ~. R. G. Buschma, "Fiboacci Numbers, Chebyshev olyomials, Geeralizatios ad Differece Equatios", Fiboacci Quarterly :, D e c, 9, pp aul F. Byrd, "Expasio of Aalytic Fuctios i olyomials Associated with Fiboacci Numbers", Fiboacci Quarterly, :, Feb. 9, pp aul F. Byrd, "Expasio of Aalytic Fuctios i T e r m s Ivolvig Lucas Numbers or Similar Number Sequeces", Fiboacci Quarterly :, April, 9, pp. . XXXXXXXXXXXXXXX
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