Polynômes - IMOSuisse

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Olympiades Suisses de Mathématiques

p x
p(x) = anxn+an−1xn−1+. . . +a1x+a0.
aip
an6= 0 anan= 1 p
p(x) = c p(x) = 0
x3+y3+z3−3xyz.
p(x) = anxn+an−1xn−1+. . . +a0q(x) = bnxn+bn−1xn−1+
. . . +b0p q
p(x)±q(x) = (an±bn)xn+ (an−1±bn−1)xn−1+. . . + (a0±b0).
c p c
c·p(x)=(c·an)xn+ (c·an−1)xn−1+. . . + (c·a0).
x
p(x)·q(x) = (anbn)x2n+ (anbn−1+an−1bn)x2n−1+. . . + k
X
i=0
ak−ibi!xk+. . . + (a0b0).
(x3−2x2+ 5)(2x2−3) =2x5−3x3−4x4+ 6x2+ 10x2−15
=2x5−4x4−3x3+ 16x2−15.
k ak6= 0 pdeg(p)
deg(0) = −∞
deg(p±q)≤max{deg(p),deg(q)},
deg(p·q) = deg(p) + deg(q).

(P, Q)
P(Q(x)) = x1994.
P(x) = amxm+. . . +arxr
Q(x) = bnxn+. . . +bsxsam6= 0, ar6= 0 bn6= 0 bs6= 0
m=r n =s P (Q(x))
P(Q(x)) = cmnxmn +. . . +crsxrs
cmn 6= 0 crs 6= 0
P(Q(x)) cmn =ambm
ncrs =arbr s
P(Q(x)) = x1994 P(x) =
xmQ(x) = xnP Q
m·n= 1994 1994 = 2 ·997
(m, n) = (1,1994),(2,997),(997,2) (1994,1)
x5+x+ 1
x5+x+ 1 = (x3+ax2+bx +c)(x2+rx +s).
a+r=0
b+ar +s=0
c+br +as =0
bs +cr =1
cs =1
c=s=±1
a=−1, b = 0, c = 1, r = 1, s = 1
x5+x+ 1 = (x3−x2+ 1)(x2+x+ 1).

n a0, . . . , anb0, . . . , bn
P≤n
P(ak) = bk0≤k≤n.
(ak, bk)P
(ak, bk)P
Lk(x) = Y
i6=k
x−ai
ak−ai
,
i0, . . . , n i =k
Lk(al) = 0 k6=l Lk(ak) = 1
P(x) =
n
X
k=0
bk·Lk(x)
Lk
ak
P ak, bk
P(x) = x(x+ 1)/2
P(a)∈Za∈Z
P≤n
P(k) = n+ 1
k−1
, k = 0,1, . . . , n.
P(n+ 1)
P(x) =
n
X
k=0 Y i6=kx−i
k−iP(k).

Y
i6=kn+ 1 −i
k−i=n+ 1 k·n k−1··· n−k+ 2 1·n−k −1··· 1 k−n
=(−1)n−kn+ 1
k.
P(n+ 1) =
n
X
k=0
(−1)n−k=(.
P3P(n)=2n−1n= 1,2,3,4
P(x) = ax3+bx2+cx +d
a+b+c+d=P(1) = 1
8a+ 4b+ 2c+d=P(2) = 2
27a+ 9b+ 3c+d=P(3) = 4
64a+ 16b+ 4c+d=P(4) = 8
a= 1/6b=−1/2c= 4/3d= 0
Z Q R
CKZ Q R C a∈K
a K b ∈K
ab = 1 ±1ZK=Q,R,C
Z
K=Z
p, q ∈K[x]
a∈K p =a·q
p, q ∈K[x]p q K
a∈K[x]p=a·q p ∈K[x]