MATHEMATICS ENCYCLOPEDIA

Using principle of mathematical induction prove that 2 n > n for all positive integers n. Let P ( n ): 2 n > n When n = 1, 2 1 > 1. Hence P ( 1 ) is true. Assume P ( k ) is true for any positive integer k, then 2 k > k                                 …( 1 ) Multiply both side of equation ( 1 ) by 2, we get 2.2 k > 2k 2 k + 1 > 2k 2 k + 1 > k + k 2 k + 1 > k + 1. Therefore P ( k + 1 ) is true when P ( k ) is true. Hence by the principle of mathematical induction, P ( n ) is true for all n.

If p ( x ) and g ( x ) are two polynomials such that g ( x ) ≠   0, then you can find polynomials r ( x ) and q ( x )   that are related as following: p ( x ) = g ( x ) ✕ q ( x ) + r ( x ) Where r ( x ) = 0 or degree of r ( x ) < degree of g ( x ) This result is known as Euclid’s Division Algorithm for polynomials. Example: Let p ( x ) = x 3 – 3x 2 + 5x – 3 and g ( x ) = x 2 – 2 Now divide p ( x ) by g ( x ) You will get quotient as q ( x ) = x – 3 and remainder as r ( x ) = 7x - 9