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

AI Thread Summary
The discussion revolves around finding the value of $f(1)$ for a polynomial $f(x)$ with integer coefficients less than 4, given that $f(4) = 2009$. Participants confirm that the polynomial's coefficients must be integers in the range of 0 to 3. The calculations reveal that the correct result for $f(1)$ is 11. The method used by a participant named kaliprasad is acknowledged as correct and effective. The focus remains on the polynomial's constraints and the derived values.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
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 $f(1)$.
 
Mathematics news on Phys.org
anemone said:
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

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
 
kaliprasad said:
= 1 + 3 + 3 + 1 + 2 + 1 = 11

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!
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top