EQUATIONS WHOSE ROOTS ARE T I E nth POWERS
OF T I E ROOTS OF A GIVEN CUBIC EQUATION
N . A , DRAiMand MARJORIE BICKNELL
Ventura, C a l i f , and Wilcox High School, Santa Clara, Calif.
Given the cubic equation
x3 - c ^ 2 + c2x - c 3 = 0
with r o o t s r l 5
._
(1)
r2,
r3j
,
the p r o b l e m of this p a p e r i s to w r i t e the equation
o
, n
n
IK o
, n n
nn
n IK
x6 - (ri + r 2 + r 3 )x* + ( r ^ + r ^ g + r 2 r 3 )x =
whose r o o t s a r e r ^ ,
of the coefficients
r2,
cl5
x3
™C(l?n)x2
+
n n n
r ^ ^
C
(2sn)X~C(35n)
=
°
r 3 , and whose coefficients a r e e x p r e s s e d in t e r m s
c2,
c3s
of the given equation 0
T h i s p a p e r extends to the cubic equation a study initiated by the solution
of a s i m i l a r p r o b l e m for the q u a d r a t i c by the s a m e a u t h o r s [ 1 1 . J u s t a s a
s p e c i a l q u a d r a t i c equation l e a d s to a r e l a t i o n s h i p between the n
Fibonacci
n u m b e r and a s u m of binomial coefficients, so does a s p e c i a l cubic equation
th
r e l a t e the n
m e m b e r of a Tribonacci sequence to a s u m of p r o d u c t s of b i nomial coefficients. Some L u c a s identities also follow.
The s u m m a t i o n for initial values of p o w e r s of r o o t s by e l e m e n t a r y t h e o r y
y i e l d s the f i r s t five e n t r i e s in the table below.
Examination of this sequence
r e v e a l s an i t e r a t i v e p a t t e r n ; namely, that if
c
(ljn)
=
r
i + r2 + r 3 ?
n > 0
s
then
C
(l,n)
=
C C
i (l9n-l) "
C
2C(l3n-2)
+
C
3C(l3n-3)
J
which i s easily p r o v e d , s i n c e each r o o t , and hence s u m s of the r o o t s , s a t i s f i e s
the original equation,,
267
268
EQUATIONS WHOSE ROOTS ARE THE nth POWERS
Sums through eighth p o w e r s appear in the table below.
[Oct.
The r i g h t - h a n d
column gives the s u m of the absolute v a l u e s of the coefficients for each value
of n.
a>
n
c
d
n)
=
r
n
i
+ r2
n
r
+
n
Coefficient
Sums for n
3
0
3
3
1
c
1
l
2
C? -
2c 2
3
cl
-
30^2
4
*\
-
4c
5
6
cf c? -
7
c7i
8
°\ -
3
i c 2 + 2c2
50^2 + 5°lc2
6cfc2 +
+ 3c 3
7
+ 40^3
11
+ 5 0 ^ 3 - 5C2C3
21
9c 2 lC 2 - 2 c | + 6c^c 3 -
+ 3c 3 2
120^203
39
- 7c5c 2 + Ucfcjj - 7 0 ^ 1 + 7cjc 3 - 21c5c 2 c 3 + 7c \cz + lci cr 23
71
8cfc2 + 20c^c| - 16cfc| + 2c£
+
8c
131
l c 3 ~ oZC-jC2Co + 2 4 c j C ^ + 1 2 c ^ c | -" 8c 2 c|
It i s p o s s i b l e to p e r c e i v e the g e n e r a l i z e d number p a t t e r n for s u m s of n
powers
of the r o o t s by extending the table above and b r e a k i n g down the s u m in t e r m s of
coefficients of p o w e r s of c 3 .
If ip
r
i s the coefficient of c 3 / n !
n
l
+
r
n
2 +
r
in the s u m
n
3
then
• 0 = c^ - nc^" 2 c 2 + n(n - 3 ) c f " 4 c | / 2 !
ip1 = n c j "
- n(n - 4)(n - 5 ) c ^ " 6 c | / 3 ! + . . .
- n(n - 4 ) ^ " c 2 + n(n - 5)(n - 6)c1~
cf/21
- n(n - 6)(n - 7)(n - 8 ) c f " 9 c f / 3 ! + • - ip2 = n(n - 5)cf* 6 - n(n - 6)(n - 7 ) c ? ' 8 c 2 " + n(n - 7)(n - 8)(n - 9)c?~ i 0 cjj/2!
- n(n - 8)(n - 9)(n - 10)(n - l l ) c f ~ 1 2 c ^ / 3 ! + - . .
1967]
O F THE ROOTS O F A GIVEN CUBIC EQUATION
leading to the t h r e e equivalent e x p r e s s i o n s below 0
269
F o r CjCgCg ^ 0,
[n/3j
(2)
r ? + r f + r3
n
=
£
c3%k/k!
,
k=o
^k
[(n-3k)/2]
\~^
, ^ m n-2m-3k m .
_.
.,.. ..
_ rt. .,
=
2^
t" 1 ) c i
c 2 n(n - m - 2k - l ) ! / ( n - 2m - 3k)!m; ,
m=o
[n/3J [ (n-3k)/2
(3)
r? + rf +
r3n
n
(4)
T1
= J]
^
k=o
m=o
(-1) n(n - m - 2k - 1)1 n-2m-3k m k
(n - 2m - 3k)Jm!k! C l
°2 ° 3
[n/3][(n-3k)/2]
=
+ r 2 + r 33 =
V
V*
2Lt
2^
k=o
(-1) n
/ n-m-2k-l\/n-m-3k\
n - m - 3k \
m=o
k
) \
Ky
X
where
[n]
i s the g r e a t e s t i n t e g e r ^ n
and
I
c
m
)
n-2m-3k m k
i
c2 c3 ,
J i s a binomial coefficient.
Notice that the coefficients of c$ a r e the s a m e as the coefficients which a r o s e
in studying the r o o t s of the q u a d r a t i c in [ 1 J .
the t e r m s
cn
.
The r e i t e r a t i v e r e l a t i o n s h i p of
s u g g e s t s a proof by m a t h e m a t i c a l induction for the t h r e e
f o r m u l a e l i s t e d , and such a proof has been w r i t t e n by the a u t h o r s .
s a k e of b r e v i t y , the proof i s omitted.
F o r the
A d e r i v a t i o n of the above f o r m u l a s
could a l s o be done using W a r i n g ' s formula and Newton's identities (see [ 2 ] ) .
Thus f a r , we have found a way to e x p r e s s the coefficient for x 2 in our
general problem.
The coefficient for
C
has a s i m i l a r computation.
(2,n)
=
x,
n n
riI>2
+
n n
riI>3
+
n n
r2r3
'
In the auxiliary cubic equation
,
n nw
n nw
n n.
_
(x - rir2 )(x - rtT3 )(x - r 2 r 3 ) = 0
notice that
c.
. is the coefficient of
(z, n;
b e c o m e s , upon multiplication,
x2.
When n = 1,
the above cubic
EQUATIONS WHOSE ROOTS ARE THE n t h
270
POWERS
[Oct.
x 3 - ( r ^ + r 1 r 3 + r 2 r 3 )x 2 + (rl|r 2 r 3 + r 1 r ^ r 3 + r 1 r 2 r | ) x - r ^ r ^ r |
=
X3 -
C 2 X 2 + CjCgX -
C§
=
0.
C o m p a r i n g this equation with the equations of our original p r o b l e m , we can
apply
the t h r e e formulae a l r e a d yJ derived for cM
to find c / 0 s if we r e ^ J
( l , n )v
(2,n)
p l a c e c 1 by c 2 ? c 2 by c^Cg, and c 3 by c | . F o r e x a m p l e , from Equation (4),
if c ^ C g fi 0,
<K
(5)
n n
vv
rtr2
our f o r m u l a for c , 2
n n
n
n
+ vr + r r
+ r i r 3 + r2r3
-
. becomes
[n/3] [(n-3k)/2]
V
V
(-1) n(n - m - 2k - 1)!
^
Z,
^ n ~ ^ n ~ ^ 3 k ) ! m! k!
k=o
m=o
m n-2m-3k 2k+m
In p r a c t i c e , when r a i s i n g the r o o t s of a given equation, It is s i m p l e r to utilize
the method of i t e r a t i n g functions than to substitute into the f o r m u l a e , especially
as n becomes larger.
An example worked by each method follows.
Given the equation
x 3 - 6x2 + l l x - 6
=
0 ,
w r i t e the cubic whose r o o t s a r e the fourth p o w e r s of the r o o t s of the given e q u a tion, without solving for the r o o t s .
(A) By substitution;
F r o m the table given e a r l i e r or from Equations (3)
and (5), the d e s i r e d cubic Is
x 3 - (cf - 4CjC2 + 2c 2 + 4c 1 c 3 )x 2 + (e2 - 4 0 ^ 0 ^ + 2G\C\ + 4c 2 c 3 )x - c 3 = 0.
Substituting
c
i = 6,
c2 = 1 1 , c3 = 6
yields
x 3 - 98x 2 + 1393x - 6 4 = 0
1967]
OF THE ROOTS O F A GIVEN CUBIC EQUATION
with r o o t s
l4,
24,
271
and 3 4 . A s a check, the r o o t s of the given equation a r e
1, 2, and 3.
(BJ By i t e r a t i o n : To get c.
C
(l,0)'
c
(l,l)'
C
i =
. , we wish to w r i t e the sequence
C
(l,3)'
C
( l , 2 ) = °21 "
2C
(l,2)'
C
(l,4)'
Now
C
(l,0)
=
3
'
C
(l,l)
=
6
>
C
2 = 36 - 22 = 14
By the i t e r a t i o n r e l a t i o n s h i p ,
3
(1,3)
=
C C
i (l52)-C2C(l,l)
C
+
(l 4)=
^
( l 5
6(6)
0)
~ n(14)
+
6
(36)
Similarly,
c / o m = 3,
'(2,0) " "'
c / 0 1N = 1 1 , c / 0 Q^ = cr22 - 2 0 ^ 3 = 49.
"(2,1) " " '
"(2,2)
Since
C
(n,2)
=
C2C
( 2 , n - l ) ~ C * C 3 c (2 5 n-2)
3)
= 251 and c
+
C
3c(2,n-3)
substitution y i e l d s
c
^
= 1393 ,
yielding the s a m e cubic as in (A).
Next let us t u r n our attention to s e v e r a l special cubic equations.
for the cubic
x3 - x2 - x - 1 -
0,
c(lj0)
-
3,
c(ljl)
-
1,
c(l^2)
First,
EQUATIONS WHOSE ROOTS ARE THE n t h POWERS.
272
[oct.
and oux r e p e a t i n g m u l t i p l i e r s in the i t e r a t i v e r e l a t i o n s h i p a r e 1, 1, 1.
c
=
(l,3)
1(3)
+
1(1)
'"'
+
1(3)
C
=
7
=
(l,n)
'
C
=
°(1 4)
(l,n-1)
+
7
C
+
3
+
(l,n-2)
X
+ C
=
U
Then,
'
(l,n-3)-
F o r this p a r t i c u l a r equation we have a s p e c i e s of T r i b o n a c c i n u m b e r s , any t e r m
after the t h i r d being the s u m of the t h r e e p r e c e d i n g t e r m s , with the entry t e r m s
3,1,3.
By Equation (4), the n
[n/3] [(n-3k)/2]
n
i—J
JL-J
k=o
m=o
term
(-l)mn
n - m
m -3k
T
\
in this T r i b o n a c c i sequence i s
/ n - m - 2 k - l \ / n - m - 3 k \
k
/\
m
/
Notice that the s u m s of the coefficients in the table given for. c,., . a r e t h e s e
(l,n)
s a m e n u m b e r s . It is i n t e r e s t i n g to r e c a l l that the special equation
x2 - x - 1 = 0
led to a f o r m u l a r e l a t i n g the n
m e m b e r of the Fibonacci sequence to a s u m
of binomial coefficients in the e a r l i e r study of the q u a d r a t i c equation [ 1 ] ,
Considering the s p e c i a l equation
x3-x
with r o o t s
+ x - l
= 0
1, ±V^T, we can w r i t e the following from Equation (4):
[ n / 3 ] [(n-3k)/2]
Zu
k=o
2
Z.
m=o
n -
m - 3k ^
k
H
'
x
m
I l if
n
.f
Q
I
'
L , defined by
J
n'
=
'
Of m o r e i n t e r e s t , however, a r e the following identities for the n
number
is odd
Lucas
1967]
O F THE ROOTS O F A GIVEN CUBIC EQUATION
Li = 1,
L 2 = 3,
Ln = ~Ln-i + L n _ 2
273
.
We substitute in Equation (3), using
;ri
= a
and letting r 3 v a r y .
=
(1 + N / 5 ) / 2 ,
r2
=
p
=
(1 -
N/5)/2
,
If r 3 = 1, Equation (3) cannot be used directly b e c a u s e
c 2 = a/3 + (3 + a = 0,
and 0° i s not defined,
s e e n that, if c 2 = 0,
/ofX
( 3!)
c ^ g £ 09
n
r,
But, by following the derivation for Equation (3), it i s
n
+
r2
[n/3]
\-^
n
+
r3 =
n(n - 2k - 1)1 n-3k k
£
(n - 3k)!k!
C
*
°*
k=o
Since
c 1 = a + fi + 1 = 2,
c 3 = a/3 = - 1 ,
and
L
n
= a
n
JI
+ Bp
substitution gives
K3]
L
n
+
X
LJ
.
k n-3k
(-1) 2
n(n - 2k - 1)]
(n - 3k)I k!
k=o
In g e n e r a l , if
r 3 = P,
P ^ 1,
P £ -1,
P ^ 0 ,
274
EQUATIONS WHOSE ROOTS ARE THE n
POWERS
OF THE ROOTS O F A GIVEN CUBIC EQUATION
Oct. 1967
Equation (3) gives
+ nn -
T
n
+
p
[n/3]
V
2-J
[(n-3k)/ 2 ]
V
m + k
(-1)
2-J
k=o
. ^ m - s k ,
^mk
n(n - m - 2k - 1)1 (p + 1)
(p-1) p
(n - 2m - 3k>! m* k!
m=o
S i m i l a r l y , Equation (31) gives the following two identities using
rt
= a,
r2
=
p,
r3=
- I/N/5" ,
and the known r e l a t i o n s h i p for Fibonacci n u m b e r s ,
F n = {an-/3n)/M5
Below,
n i s taken to be 2s + 1 and 2s
F 92S+1 ~
1/58+1
=
°
~
s+i _
X/
T
+ i / ,
L 2 g + 1/5
s
_
-
.
respectively.
f ( 2 S + l ) / 3 J ,n
,w„
^k,2S-3k + i
v-*
(2s + 1) (2s - 2k) ! (-1) 4
E
/_j
/rt
—Q
[2s/3J
V
>
^
_
s-k+i
(2s - 3k + l ) ! k ! 5 ^
k
2s(2s
- 2k - 1) I (-if
:
—£
(2s - 3k)J k! 5 S K
? s-3k
42S 3 k
REFERENCES
1.
N. A. D r a i m and M, Bicknell, "Sums of the n
Given Q u a d r a t i c Equation,
M
P o w e r s of the Roots of a
The Fibonacci Q u a r t e r l y , 4:2, A p r i l , 1966y
pp. 170-178.
2.
W. S. B u r n s i d e and A. W. Panton, An Introduction to the T h e o r y of Binary
A l g e b r a i c F o r m s , Dover, New York, 1960, Chapter 15.
• • * * •