GATE CS 1997 Q4.4 — Sets, Relations and Functions

We can determine the minimum degree of the polynomial $p(x)$ using the method of finite differences for the equally spaced points $x = 0, 1, 2, 3$.

Let's construct the difference table:

Since the second-order differences are constant (equal to $6$), the polynomial $p(x)$ has a minimum degree of $2$.

Verification:
Let $p(x) = ax^2 + bx + c$. Using the given values:

• $p(0) = c = 5$
• $p(1) = a + b + 5 = 4 \implies a + b = -1$
• $p(2) = 4a + 2b + 5 = 9 \implies 2a + b = 2$

Solving these equations gives $a = 3$ and $b = -4$.
Thus, the polynomial is $p(x) = 3x^2 - 4x + 5$.
Checking for $x = 3$: $p(3) = 3(3)^2 - 4(3) + 5 = 27 - 12 + 5 = 20$, which is consistent with the given data.

Therefore, the minimum degree is $2$.