Question

The expression $ {(1 + x)^n} - nx - 1;n \in \mathbb{N} $ is divisible by which of these
A. $ 2x $
B. $ {x^2} $
C. $ 2{x^3} $
D. All of these

Answer 1

Hint: In this question, we put the value of $ n $ and use hit and trial to see which option divides the given algebraic expression. We can use this method to solve the question in a quick time. For an elaborated answer we must expand the binomial expression.

Complete step-by-step answer:
Firstly we write the expression and see how to proceed
 $ {(1 + x)^n} - nx - 1;n \in \mathbb{N} $
This expression is divisible by one of the given options, so must try to solve it by hit and trial method and put values of ‘n’ to check for the options
As we know that ‘n’ is a natural number so we try to solve this way
 $
  n = 1 \\
   \Rightarrow {(1 + x)^1} - 1 \times x - 1 = 1 + x - x - 1 = 0 \;
  $
So zero is divisible by everything so we have to check for other number so we take
 $
  n = 2 \\
   \Rightarrow {(1 + x)^2} - 2x - 1 \;
  $
Using the formula of binomial expansion i.e.
 $ {(1 + x)^2} = 1 + {x^2} + 2x $
Putting the value in the above equation
 $
  {(1 + x)^2} - 2x - 1 = 1 + {x^2} + 2x - 2x - 1 \\
   \Rightarrow {x^2} \;
  $
So the result comes out to be $ {x^2} $ so our option ‘b’ is the correct answer.
We also check for $ n = 3 $ to see if our result is same or not i.e.
 $
  {(1 + x)^3} - 3x - 1 \\
   \Rightarrow {(1 + x)^3} = 1 + {x^3} + 3x + 3{x^2} \\
   \Rightarrow 1 + {x^3} + 3x + 3{x^2} - 3x - 1 \;
  $
So after cancellation of the terms we are left with
 $ {x^3} + 3{x^2} $
Taking $ {x^2} $ common from the expression we have
 $ {x^2} \times (x + 3) $
We can clearly see that from all our options only option ‘b’ which is $ {x^2} $ , divides the expression.
Hence our answer is $ {x^2} $ .
This is the way we check for the options and get to our answer.
So, the correct answer is “ $ {x^2} $ ”.

Note: Hit and trial methods can be very useful while solving such problems where putting values can simplify the question and we can solve problems in a short time. Options also help us here to find the solution because for an elaborated solution we have to use the binomial theorem to obtain our result.