Maths-
General
Easy

Question

Use the binomial theorem to expand the expressions:
open parentheses straight m squared plus straight n over 2 close parentheses cubed

The correct answer is: (m^2+n/2 )^3 = m^6 + (3m^4 n)/2 + (3m^2 n^2)/4 + n^3/8


    ANSWER:
    Hint:
    The binomial expansion is 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   nC subscript straight k straight x to the power of straight n minus straight k end exponent straight y to the power of straight k comma text  here  end text straight n greater or equal than 0,
    We are asked to find the expansion of  open parentheses straight m squared plus straight n over 2 close parentheses cubed using Binomial theorem.
    Step 1 of 2:
    The given expression isopen parentheses straight m squared plus straight n over 2 close parentheses cubed. Here, the value of n = 3 so there are 3+1 = 4 terms in the expansion of the expression. The values of  straight x equals straight m squared straight & straight y equals straight n over 2.
    Step 2 of 2:
    Substitute the values in the binomial equation to get the expansion;
    table attributes columnalign right columnspacing 0em end attributes row cell open parentheses straight m squared plus straight n over 2 close parentheses cubed equals 3 straight C subscript 0 open parentheses straight m squared close parentheses cubed plus 3 straight C subscript 1 open parentheses straight m squared close parentheses squared open parentheses straight n over 2 close parentheses plus 3 straight C subscript 2 open parentheses straight m squared close parentheses open parentheses straight n over 2 close parentheses squared plus 3 straight C subscript 3 open parentheses straight n over 2 close parentheses cubed end cell row cell equals straight m to the power of 6 plus 3 open parentheses straight m to the power of 4 close parentheses open parentheses straight n over 2 close parentheses plus 3 open parentheses straight m squared close parentheses open parentheses straight n squared over 4 close parentheses plus straight n cubed over 8 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end cell row cell equals straight m to the power of 6 plus fraction numerator 3 straight m to the power of 4 straight n over denominator 2 end fraction plus fraction numerator 3 straight m squared straight n squared over denominator 4 end fraction plus straight n cubed over 8 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end cell end table
    Thus, the expansion is: open parentheses straight m squared plus straight n over 2 close parentheses cubed equals straight m to the power of 6 plus fraction numerator 3 straight m to the power of 4 straight n over denominator 2 end fraction plus fraction numerator 3 straight m squared straight n squared over denominator 4 end fraction plus straight n cubed over 8
    Note:
    The answer can also be found using the Pascal’s triangle. For the expansion of the expression (x+y)n , we would consider the (n+1)th row in the triangle.

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