[superprismatic]

I'm thinking of a polynomial P(x) which

has non-negative integer coefficients.

Now, P(1)=14 and P(15)=5296846548.

What is the polynomial, P(x), of which

I think?

Please use spoilers.

[araver]

Let

P(x)=x^n*a_n+...+x*a_1+a_0 

with a_i>=0 (non-negative). Then P(1)=Sum a_i = 14 means there are at most 14 positive coefficients a_i (but n can still be more than 14) and at least 1 positive coefficient (otherwise P(1)=0). Note that 0<=a_i<=14 for all i. P(15)=5296846548 so P(15) mod 2 = 0, so there is an even number of positive coefficients. P(15) mod 15 = 3 so a_0=3 (mod 15) so a_0=3. Let Q1(x)=(P(x)-3)/x. Then Q1(1)=14-3=11, Q1(15)=353123103. Again Q1(15) mod 15 = 3 so a_1 =3 (mod 15) so a_1=3. Let Q2(x)=(Q1(x)-3)/x. Then Q2(1)=11-3=8 and Q2(15)=23541540 Q2(15) mod 15 = 0 so a_2=0 (mod 15) so a_2=0 And so on. What we're doing is basically converting 5296846548 in base 15 to obtain the coefficients of P. Since 0<=a_i<=14 for all i, they are all valid digits in base 15. 5296846548 (base15)=210041033 which means

P(x)=2*x^8+x^7+4*x^4+x^3+3*x+3

[dark_magician_92]

I'm thinking of a polynomial P(x) which

has non-negative integer coefficients.

Now, P(1)=14 and P(15)=5296846548.

What is the polynomial, P(x), of which

I think?

Please use spoilers.

some hints plz.

[superprismatic]

some hints plz.

What does the value of P(1) tell you about the polynomial?

Specifically, what does it tell you about the coefficients?

I'll post a new hint every few days. Thanks for asking.

[Guest]

PolyNomial?Who invented that word?

[Guest]

Y = 2x^8 + x^7 + 4x^4 + x^3 + 3x + 3

Since there are no negative coefficients, and all coefficients sum to 14 (i.e. f(1) = 14), there can be no coeffecients greater than 14. So this is really like asking how do you represent this big number in base 15, and do the base 15 digits sum to 14. It turns out that they do:

5296846548 (base 10) = 210041033 (base 15)

To check this, convert the given base 15 number to base 10, remembering that each succesive place is just the next power of 15; 15^0, 15^1, 15^2, etc.

[dark_magician_92]

I'm thinking of a polynomial P(x) which

has non-negative integer coefficients.

Now, P(1)=14 and P(15)=5296846548.

What is the polynomial, P(x), of which

I think?

Please use spoilers.

f(x)= 2(x^8)+(x^7)+4(x^4)+x^3+3x+3

[superprismatic]

araver, bartf, and dark_magician_92 all solved it.