# Method of differences

Hello everyone ,

I've just learnt the "Method of differences" and I'm really fascinated by the method !

Can somebody help me out how can we find the original polynomial $f(x)$ ,

[ For example $f(x)$ of the form $f(x)= ax^n + bx^{n-1} + \dots$ ]

after drawing the difference table (i.e. after finding the values of $f(a) , D_1(a) \dots$ ) ?

Do elaborate 'cause I don't know much about Binomials .

Thank you. Note by Priyansh Sangule
6 years, 4 months ago

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Are you talking about the values of $f(n)$, or the closed form?

- 6 years, 4 months ago

The original polynomial of the form $f(x) = ax^n + bx^{n-1} + \dots$

- 6 years, 4 months ago

Okay. Could you give an example of such a table? I'd be happy to help you find the original polynomial.

- 6 years, 4 months ago

For example -

A cubic polynomial $f(x)$ has the following values -

$f(1)=13 ; f(2)=32 ; f(3)=69 ; f(4)=130$

Find $f(x)$ .

So for this , first of all , I draw a difference table

$\begin{matrix} n & f(n) & D_1(n) & D_2(n) & D_3(n) & \dots \\ 1 & 13 & 19 & 18 & 6 & \dots \\ 2 & 32 & 37 & 24 & \dots \\ 3 & 69 & 61 & \dots \\ 4 & 130 & \dots \\ \vdots \end{matrix}$

Now , at this point I run into trouble - that how can I find the polynomial of the form $f(x) = ax^n + bx^{n-1} + \dots$

Thank you

- 6 years, 4 months ago

I'm encourage you to read through Worked Example 1, and do the (*) exercise, which gives you the answer.

Staff - 6 years, 4 months ago

Yes sir, I did note that but couldn't it be written in a simpler form ? I saw some people using combinatorics for that!

- 6 years, 4 months ago

I need help guys :)

- 5 years, 6 months ago