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Binomial Theorem And Its Simple Applications

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

Binomial Theorem

Key Concepts

  1. A binomial expression is an algebraic expression having two terms. For example, (a + b). (a - b) etc.
  2. The expansion of a binomial for any positive integral exponent W is given by the binomial theorem. The binomial theorem says that (x+ y)n = nC1xn-1y+ nC2xn-2y2+ --- + nCrxn-ryr + ----- + nCn-1xyn-1 + nCnyn In summation notation begin mathsize 12px style left parenthesis straight x plus straight y right parenthesis to the power of straight n equals sum from straight k equals 0 to straight n of to the power of straight n straight C subscript straight k straight x to the power of straight n minus straight k space end exponent straight y to the power of space straight k end exponent end style
    • In the binomial expansion of (x + y)n the number of terms is (n + 1) i.e. one more than the exponent.
    • The exponent of 'I goes on decreasing by unity and that of y increases by unity. Exponent of 'x is W in the first term, (n - 1) in the second term. and so on ending with zero in the last term.
    • The sum of the indices of 'x and 'y' is always equal to the index of the expression.
  3. The coefficients nCr. the number of combinations of n objects taken rata time. Occurring in the binomial theorem, are known as binomial coefficients.
  4. Binomial coefficients when arranged in the form given below form the Pascal's Triangle 
  5. The array of numbers arranged in the from of a triangle with 1 at the vertes and running down two slanting sides is known as the Pascal’s triangle, after the name of the French mathematician Blaise Pascal. It is also known as Menu Prastara by Pinglak.
  6. Pascal's triangle is a special triangle of numbers. It has an infinite number of rows. Pascal's triangle is a storehouse of patterns.
  7. In order to construct the elements of the following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place.
  8. Using the binomial theorem for non -negative index
    (x-y)n = [x +(-y)]n
    (x-y)n= nCoXn - nC1Xn-1y + nC2Xn-2y2  - nC3Xn-3y3 + ….+ (-1)n  nCnyn 
    In summation notation begin mathsize 12px style left parenthesis straight x minus straight Y right parenthesis to the power of straight n equals sum from straight k minus 0 to straight n of left parenthesis negative 1 right parenthesis to the power of straight k text   end text to the power of straight n text C end text subscript straight k straight x to the power of straight n minus straight k text   end text end exponent straight y text   end text to the power of straight k end style
  9. The binomial theorem can be used to expand a trinomial by applying the binomial expansion twice.
  10. General term in the expansion of  begin mathsize 12px style left parenthesis straight x plus straight y right parenthesis to the power of straight n text  is T end text subscript straight k plus 1 end subscript equals to the power of straight n straight C subscript straight k text  x end text to the power of straight n minus straight k end exponent text   end text straight y to the power of space straight k end exponent end style
  11. General term in the expansion of begin mathsize 12px style left parenthesis straight x plus straight y right parenthesis to the power of straight n text  is T end text subscript straight k plus 1 end subscript equals to the power of straight n straight C subscript straight k text  x end text to the power of straight n minus straight k end exponent text   end text straight y space to the power of straight k end style
  12. If n is odd, then {(x + y)n + (x - y)"} and {(x + y)n - (x - y)n} both have the same number of terms equal to begin mathsize 12px style fraction numerator straight n plus 1 over denominator 2 end fraction end style  whereas if n is even, then {(x + y)n + (x - y)n} has  begin mathsize 12px style left parenthesis straight n over 2 plus 1 right parenthesis to the power of th end style terms and begin mathsize 12px style left curly bracket left parenthesis straight x plus straight y right parenthesis to the power of straight n minus left parenthesis straight x minus straight y right parenthesis to the power of straight n right curly bracket end style has begin mathsize 12px style left parenthesis straight n over 2 right parenthesis end style  terms.
  13. If n is even , there is only one middle term in the expansion of (x + y)n and will be the begin mathsize 12px style left parenthesis straight n over 2 plus 1 right parenthesis to the power of th end style term
  14. If n is odd, there are two middle terms in the expansion of  begin mathsize 12px style left parenthesis straight x plus straight y right parenthesis to the power of straight n comma straight i. straight e. the text   end text left parenthesis fraction numerator straight n plus 1 over denominator 2 end fraction right parenthesis to the power of th and text   end text left parenthesis fraction numerator straight n plus 3 over denominator 2 end fraction right parenthesis to the power of th text  term. end text end style

 

 

 

 

 

 

 

 

 

 

 

 

 

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