Finding $f(1)$ in a Polynomial of Integer Coefficients $\leq$ 4

[anemone]

Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$.

Given that $f(4)=2009$, find $f(1)$.

 

[kaliprasad]

Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$.

Given that $f(4)=2009$, find $f(1)$.

= 1 + 3 + 3 + 1 + 2 + 1 = 11

Spoiler

as f(x) = x^5 + 3x^4 + 3x^3 + x^2 +2x +1

as no coefficient is >4 and we are given f(4) subtract the highest power of 4 as many times as it can go

2009 = 1024 + 985
985 = 256 * 3 + 217
217 = 64 * 3 + 25
25 = 16 + 9
9 = 2 *4 + 1

 

[anemone]

= 1 + 3 + 3 + 1 + 2 + 1 = 11

Spoiler

as f(x) = x^5 + 3x^4 + 3x^3 + x^2 +2x +1

as no coefficient is >4 and we are given f(4) subtract the highest power of 4 as many times as it can go

2009 = 1024 + 985
985 = 256 * 3 + 217
217 = 64 * 3 + 25
25 = 16 + 9
9 = 2 *4 + 1

Hey kaliprasad,

Thanks for participating and yes, the answer is correct and your method is great and nice!