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Binomial Theorem and its Simple Applications

Bionomial Theorem and its Simple Applications PDF Notes, Important Questions and Synopsis

SYNOPSIS

  1. A binomial is a polynomial having only two terms. For e.g 2y2- 1
  2. (x + y)n can be expanded using the Binomial theorem without actually multiplying it n times.
    begin mathsize 12px style left parenthesis straight x plus straight y right parenthesis to the power of straight n equals straight C presuperscript straight n subscript 0 straight x to the power of straight n straight y to the power of 0 plus straight C presuperscript straight n subscript 1 straight x to the power of straight n minus 1 end exponent straight y to the power of 1 plus straight C presuperscript straight n subscript 2 straight x to the power of straight n minus 2 end exponent straight y squared plus. ... plus straight C presuperscript straight n subscript straight n straight x to the power of 0 straight y to the power of straight n equals sum from straight r equals 0 to straight n of straight C presuperscript straight n subscript straight r straight x to the power of straight n minus straight r end exponent straight y to the power of straight r end style
  3. Properties of Binomial Expansion (x + y)n

    i.Total number of terms in this expansion is n + 1.

    ii.The exponent of x decreases by 1, while the exponent of y increases by 1 in subsequent terms.

    iii.The first term is  begin mathsize 12px style straight C presuperscript straight n subscript 0 straight x to the power of straight n straight y to the power of 0 equals straight x to the power of straight n end style and the final/last term is  begin mathsize 12px style straight C presuperscript straight n subscript straight n straight x to the power of 0 straight y to the power of straight n equals straight y to the power of straight n. end style

    iv.The general term in this expansion is given by
     begin mathsize 12px style straight T subscript straight r plus 1 end subscript equals straight C presuperscript straight n subscript straight r straight x to the power of straight n minus straight r end exponent straight y to the power of straight r end style

  4. Binomial Coefficients:
    Binomial coefficients in the expansion of (x + y)n are simply the number of ways of choosing x from the brackets and y from the rest.

  5. Pascal’s Triangle:
    Binomial coefficients can be found using Pascal’s triangle given below.

    We can also say that
    begin mathsize 12px style straight C presuperscript straight n subscript straight r plus straight C presuperscript straight n subscript straight r minus 1 end subscript equals straight C presuperscript straight n plus 1 end presuperscript subscript straight r end style.
    Also, the binomial coefficient is given by 
     begin mathsize 12px style straight C presuperscript straight n subscript straight r equals fraction numerator straight n factorial over denominator straight r factorial left parenthesis straight n minus straight r right parenthesis factorial end fraction end style

  6. Terms in the binomial expansion of (x + y)n

    begin mathsize 12px style straight T subscript straight r plus 1 end subscript equals straight C presuperscript straight n subscript straight r straight x to the power of straight n minus straight r end exponent straight y to the power of straight r end style
    i.General Term (rth term): General term in the expansion of (x + y)n is given by

    ii.Middle Term(s):

       1. When n is even,
           begin mathsize 12px style straight T subscript fraction numerator straight n plus 2 over denominator 2 end fraction end subscript equals straight C presuperscript straight n subscript straight n over 2 end subscript straight x to the power of straight n over 2 end exponent times straight y to the power of straight n over 2 end exponent end style

      2. When n is odd,
        begin mathsize 12px style straight T subscript fraction numerator straight n plus 1 over denominator 2 end fraction end subscript end style and begin mathsize 12px style straight T subscript fraction numerator straight n plus 3 over denominator 2 end fraction end subscript end style

    iii.Greatest Term:
    In any binomial expansion, the values of the terms increase, reach a maximum and then decrease.

    So, to find the greatest term, find the value of r till  begin mathsize 12px style straight T subscript straight r plus 1 end subscript over straight T subscript straight r greater or equal than 1 end style
    begin mathsize 12px style rightwards double arrow fraction numerator straight n minus straight r plus 1 over denominator straight r end fraction straight x greater or equal than 1 rightwards double arrow straight r less or equal than fraction numerator left parenthesis straight n plus 1 right parenthesis straight x over denominator 1 plus straight x end fraction end style
    So, the greatest term occurs when
    r = begin mathsize 12px style fraction numerator left parenthesis straight n plus 1 right parenthesis straight x over denominator 1 plus straight x end fraction end style
    iv.Term independent of x:

    The term independent of x is the term not containing x.
    So, find the value of r such that the exponent of x is zero.

  7. Applications of Binomial Expansion:
    i. We have a very important result

       begin mathsize 12px style 2 less or equal than open parentheses 1 plus 1 over straight n close parentheses to the power of straight n less than 3 end style, n ≥ 1, n ∊ N
    ii. Finding the remainder using Binomial Theorem:
    To find the remainder when pn is divided by q, adjust the power of p to pm which is very close to b, say with difference 1 and then divide by taking the remainder always positive.

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