GATE | GATE-CS-2000 | Question 49

A polynomial p(x) satisfies the following:

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

The minimum degree of such a polynomial is
(A) 1
(B) 2
(C) 3
(D) 4


Answer: (D)

Explanation:

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

The polynomial touches 0 at least once from 1 to 2, so there
is a root between 1 to 2

The polynomial touches 0 at least once from 2 to 3, so there
is a root between 2 to 3

Similarly, there is at least one root from 3 to 4 and 4 to 5.

So minimum degree is 4.


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