What is the minimum degree of a polynomial, given the initial conditions?

A polynomial $ p(x)$ is such that $p(0) =5, p(1) = 4, p(2) = 9 $ and $p(3) = 20 $.

What is the minimum degree it can have?

The problem can easily be solved by hit and trial method. First assuming that the degree is 1, then 2 and so on until the initial conditions are satisfied. This method gives the answer as 2, for the above problem.

But in spite of checking each and every value, starting from 1; is there any better approach to solve such problems ?

Many thanks !!

1 Answer

For a linear polynomial $p$, you'll always have $p(n+1) - p(n)$ the same. If you write down a table

 1 2 3 4 5
p(1) p(2) p(3) p(4) p(5)

which in your case would be this:

0 1 2 3
5 4 9 20

and then write the differences $p(2) - p(1)$, $p(3) - p(2)$, etc in a row beneath, you'd get (again in your case)

0 1 2 3
5 4 9 20 -1 5 11

That new row is called the "first differences". For a linear function, the entries would all be the same. You can also write down second differences:

0 1 2 3
5 4 9 20 -1 5 11 6 6

For a quadratic function, these second differences will all be the same. Your values actually are all the same, so your function can be expressed as a quadratic.