Question

Polynomials
How many Polynomials (with real coefficients) are there of degree 10 such that

\[P(x^2)=P(x) .P(x-1)\]

Answer 1

lets think together...
i dont have any idea

Answer 2

i think.....in p(x^2) the exponents of x are even....
but but in p(x)P(x-1) the exponents of x may be even or odd unless a specific condition is present....
and i have no clue about this condition!!!

Answer 3

i found this hint

Note first that p(1) = p(1) p(0). So p(1) = 0 or p(0) = 1. Suppose that p(1) = 0.
Then p(2^2) = p(2) p(1) = 0. Continuing in this way, we deduce that p(n) = 0 for infinitely many positive integers n. As a nonconstant polynomial, p(x) can have only finitely many roots.Thus we have a contradiction, and can conclude that p(0) = 1.
Suppose p(a) = 0 for some complex number a. Then p(a^2) = p(a^4) =· · · = 0. However, a polynomial can only have a finite number of roots. So a must be a root of unity. Therefore, |a| = 1. Similarly, we can prove that |a + 1| = 1.

Answer 4

now if we let a=x+iy with some simple calculations we get
\[a=-\frac{1}{2}\pm i \frac{\sqrt{3}}{2}\]
since |a| = 1 and |a + 1| = 1

Answer 5

then they are the only roots of a such polynomial
now i think we can say
\[P(x)=c((x+\frac{1}{2}-i \frac{\sqrt{3}}{2})(x+\frac{1}{2}+i \frac{\sqrt{3}}{2}))^n=c(x^2+x+1)^n\]

Answer 6

P(0)=1
then c=1
and 2n=10 --> n=5

so i think the only answer is
\[P(x)=(x^2+x+1)^5\]

Answer 7

@Zarkon @experimentX

Answer 8

@satellite73
plz check the solution if u can

Answer 9

That is what I got

\[
\left(x^2+x+1\right)^5
\]

Answer 10

I solve the following equations involving the coefficient using Mathematica
\[

a(0)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)-a(0)=0\\ 2 a(9)-10=0\\
2 a(7)-8 a(8)+a(8) (a(9)-10)+(a(8)-9
a(9)+45) a(9)+36 a(9)-120=0\\ 2
a(8)+(a(9)-10) a(9)-10 a(9)+45=0\\
a(1)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(0) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)=0\\ (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10) a(1)-a(1)+a(0) (a(2)-3 a(3)+6
a(4)-10 a(5)+15 a(6)-21 a(7)+28 a(8)-36
a(9)+45)+a(2)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)=0\\ 2 a(5)-6 a(6)+21
a(7)-56 a(8)+a(7) (a(8)-9 a(9)+45)+a(6)
(a(9)-10)+(a(6)-7 a(7)+28 a(8)-84
a(9)+210) a(9)+126 a(9)+a(8) (a(7)-8
a(8)+36 a(9)-120)-252=0\\ 2 a(6)-7
a(7)+27 a(8)+a(8) (a(8)-9 a(9)+45)+a(7)
(a(9)-10)-84 a(9)+a(9) (a(7)-8 a(8)+36
a(9)-120)+210=0\\ a(1) (a(2)-3 a(3)+6
a(4)-10 a(5)+15 a(6)-21 a(7)+28 a(8)-36
a(9)+45)+a(3)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(2) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(0) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)=0\\
(a(2)-3 a(3)+6 a(4)-10 a(5)+15 a(6)-21
a(7)+28 a(8)-36 a(9)+45) a(2)-a(2)+a(0)
(a(4)-5 a(5)+15 a(6)-35 a(7)+70 a(8)-126
a(9)+210)+a(4)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(3) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(1) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)=0\\
a(1) (a(4)-5 a(5)+15 a(6)-35 a(7)+70
a(8)-126 a(9)+210)+a(3) (a(2)-3 a(3)+6
a(4)-10 a(5)+15 a(6)-21 a(7)+28 a(8)-36
a(9)+45)+a(5)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(4) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(2) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)+a(0)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)=0\\ (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)
a(3)-a(3)+a(2) (a(4)-5 a(5)+15 a(6)-35
a(7)+70 a(8)-126 a(9)+210)+a(0) (a(6)-7
a(7)+28 a(8)-84 a(9)+210)+a(4) (a(2)-3
a(3)+6 a(4)-10 a(5)+15 a(6)-21 a(7)+28
a(8)-36 a(9)+45)+a(6)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(5) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(1) (a(5)-6 a(6)+21 a(7)-56
a(8)+126 a(9)-252)=0\\ a(3) (a(4)-5
a(5)+15 a(6)-35 a(7)+70 a(8)-126
a(9)+210)+a(1) (a(6)-7 a(7)+28 a(8)-84
a(9)+210)+a(5) (a(2)-3 a(3)+6 a(4)-10
a(5)+15 a(6)-21 a(7)+28 a(8)-36
a(9)+45)+a(7)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(6) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(0) (a(7)-8 a(8)+36
a(9)-120)+a(4) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)+a(2)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)=0\\ (a(4)-5 a(5)+15 a(6)-35
a(7)+70 a(8)-126 a(9)+210) a(4)-a(4)+a(2)
(a(6)-7 a(7)+28 a(8)-84 a(9)+210)+a(6)
(a(2)-3 a(3)+6 a(4)-10 a(5)+15 a(6)-21
a(7)+28 a(8)-36 a(9)+45)+a(0) (a(8)-9
a(9)+45)+a(8)
(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+a(6)-a(7)+
a(8)-a(9)+1)+a(7) (a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+9
a(9)-10)+a(1) (a(7)-8 a(8)+36
a(9)-120)+a(5) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)+a(3)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)=0\\ a(5) (a(4)-5 a(5)+15
a(6)-35 a(7)+70 a(8)-126 a(9)+210)+a(3)
(a(6)-7 a(7)+28 a(8)-84 a(9)+210)+a(7)
(a(2)-3 a(3)+6 a(4)-10 a(5)+15 a(6)-21
a(7)+28 a(8)-36 a(9)+45)+a(1) (a(8)-9
a(9)+45)+a(0)
(a(9)-10)+(a(0)-a(1)+a(2)-a(3)+a(4)-a(5)+
a(6)-a(7)+a(8)-a(9)+1) a(9)+a(8) (a(1)-2
a(2)+3 a(3)-4 a(4)+5 a(5)-6 a(6)+7 a(7)-8
a(8)+9 a(9)-10)+a(2) (a(7)-8 a(8)+36
a(9)-120)+a(6) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)+a(4)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)=0\\ 2
a(0)-a(1)+a(2)-a(3)+a(4)-2
a(5)+a(6)-a(7)+a(8)+a(6) (a(4)-5 a(5)+15
a(6)-35 a(7)+70 a(8)-126 a(9)+210)+a(4)
(a(6)-7 a(7)+28 a(8)-84 a(9)+210)+a(8)
(a(2)-3 a(3)+6 a(4)-10 a(5)+15 a(6)-21
a(7)+28 a(8)-36 a(9)+45)+a(2) (a(8)-9
a(9)+45)+a(1) (a(9)-10)-a(9)+a(9) (a(1)-2
a(2)+3 a(3)-4 a(4)+5 a(5)-6 a(6)+7 a(7)-8
a(8)+9 a(9)-10)+a(3) (a(7)-8 a(8)+36
a(9)-120)+a(7) (a(3)-4 a(4)+10 a(5)-20
a(6)+35 a(7)-56 a(8)+84 a(9)-120)+a(5)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)+1=0\\ 2 a(1)-2 a(2)+3 a(3)-4
a(4)+5 a(5)-6 a(6)+7 a(7)-8 a(8)+a(7)
(a(4)-5 a(5)+15 a(6)-35 a(7)+70 a(8)-126
a(9)+210)+a(5) (a(6)-7 a(7)+28 a(8)-84
a(9)+210)+a(3) (a(8)-9 a(9)+45)+a(2)
(a(9)-10)+(a(2)-3 a(3)+6 a(4)-10 a(5)+15
a(6)-21 a(7)+28 a(8)-36 a(9)+45) a(9)+9
a(9)+a(4) (a(7)-8 a(8)+36 a(9)-120)+a(8)
(a(3)-4 a(4)+10 a(5)-20 a(6)+35 a(7)-56
a(8)+84 a(9)-120)+a(6) (a(5)-6 a(6)+21
a(7)-56 a(8)+126 a(9)-252)-10=0\\ 2
a(2)-3 a(3)+6 a(4)-10 a(5)+14 a(6)-21
a(7)+28 a(8)+a(8) (a(4)-5 a(5)+15 a(6)-35
a(7)+70 a(8)-126 a(9)+210)+a(6) (a(6)-7
a(7)+28 a(8)-84 a(9)+210)+a(4) (a(8)-9
a(9)+45)+a(3) (a(9)-10)-36 a(9)+a(5)
(a(7)-8 a(8)+36 a(9)-120)+a(9) (a(3)-4
a(4)+10 a(5)-20 a(6)+35 a(7)-56 a(8)+84
a(9)-120)+a(7) (a(5)-6 a(6)+21 a(7)-56
a(8)+126 a(9)-252)+45=0\\ 2 a(3)-4
a(4)+10 a(5)-20 a(6)+35 a(7)-56 a(8)+a(7)
(a(6)-7 a(7)+28 a(8)-84 a(9)+210)+a(5)
(a(8)-9 a(9)+45)+a(4) (a(9)-10)+(a(4)-5
a(5)+15 a(6)-35 a(7)+70 a(8)-126
a(9)+210) a(9)+84 a(9)+a(6) (a(7)-8
a(8)+36 a(9)-120)+a(8) (a(5)-6 a(6)+21
a(7)-56 a(8)+126 a(9)-252)-120=0\\ 2
a(4)-5 a(5)+15 a(6)-36 a(7)+70 a(8)+a(8)
(a(6)-7 a(7)+28 a(8)-84 a(9)+210)+a(6)
(a(8)-9 a(9)+45)+a(5) (a(9)-10)-126
a(9)+a(7) (a(7)-8 a(8)+36 a(9)-120)+a(9)
(a(5)-6 a(6)+21 a(7)-56 a(8)+126
a(9)-252)+210=0
\]

Answer 11

oh u solved for a0,a1,a2,...
so nice
thank u @eliassaab

Answer 12

I got\[
\{\{a(0)\to 1,a(1)\to 5,a(2)\to 15,a(3)\to
30,a(4)\to 45,a(5)\to 51,a(6)\to
45,a(7)\to 30,a(8)\to 15,a(9)\to 5\}\}
\]
Which gives
\[
P(x)=
\]

Answer 13

\[
P(x)= x^{10}+5 x^9+15 x^8+30 x^7+45 x^6+\\51 x^5+45
x^4+30 x^3+15 x^2+5 x+1=\left(x^2+x+1\right)^5
\]

Answer 14

special thanks to MATHEMATICA and u
so my solution is right

Answer 15

I took a(10) =1

Answer 16

yw

Answer 17

It seems the only polynomials of 2p degress that satisfy that
P(x^2)=P(x)P(x-1) is of the from
\[
\left ( 1+x+x^2 \right)^p
\]

Answer 18

The above statement is always true.