MCQs in Computer Science - Timothy Williams

Question

The $n^{th}$ order difference of a polynomial of degree $n$ is

1. zero
2. one
3. some constant
4. undefined

Please explain the solution.

Answer 1

Answer is C, i.e. Some Constant. 

First Order Difference is nothing but First order derivative. Similarly, n^th order difference means n^th  order Derivative. 

Take polynomial f(x) of Order n : 

(Where a_i  is any Real Number and  a₀  is NOT equal to Zero.) 

First Order Difference will be : $f'(x) = a_{0}\,n\, x^{n-1} + a_1 (n-1)\, x^{n-2} + \cdot \cdot \cdot \cdot +\,\, a_{n-1}$    i.e. a polynomial of (n-1) degree.

Similarly, You can see that Second Order difference (or Derivative) will be a Polynomial of (n-2) degree. And So on, The 

$n^{th}$ degree polynomial would just be a "Zero Degree" Polynomial which is : 

Which is Some Constant. 

We can note the following :

1. The n^th difference, Δⁿ,   is n! a₀
2. The n+1 difference, Δⁿ⁺¹, and higher differences, are all  zero.

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Note That : 

Zero Degree Polynomial : $f(x) = a_0\,x^{0} = a_0$  (Where $a_0$ is any Real Number and can even be Zero) 

One Degree Polynomial : $f(x) = a_0\,x^{1} + a_1 = a_0 \,x + a_1$  (Where $a_i$ is any Real Number and $a_0$ Can NOT be Zero)

and So on.