Question

Use Principle of Mathematical induction to show that the
polynomial P(X) = X3
– X, is divisible by 6, where X is a
non-negative integer.

Answer 1

P(x) = x³ - x       is divisible by  6,      where x = non negative integer.

P(x=0) = 0  , divisible by 6
P(x=1) = 0     divisible by 6
P(x=2) = 8-2 = 6    divisible by 6

Let   P(x) be divisible by 6 for  x >= 0.
P(x+1) = (x+1)³ -  (x+1)
            = x³ + 3 x³ + 3 x + 1 - x - 1
            = (x³ - x)  +  3 x (x² + 1)      ----- (1)

If x is even, then x²+1 is an odd number.
If  x is odd, then  x² + 1 is an even number.
Hence,   x (x² + 1)  is always divisible by 2.
Hence the RHS on  (1) is always divisible by  6

Applying the mathematical induction principle,  P(x) is divisible by 6 for x >= 0.