Maths-
General
Easy

Question

Use the binomial theorem to expand the expressions:
left parenthesis 4 straight g plus 2 straight h right parenthesis to the power of 4

The correct answer is: (4g+2h)^4 = 256g^4 + 512g^3 h + 384g^2 h^2 + 128gh^3 + 16h^4


    ANSWER:
    Hint:
    The binomial expansion is left parenthesis x plus y right parenthesis to the power of n equals sum from k equals 0 to n of   n C subscript k x to the power of n minus k end exponent y to the power of k comma text  here  end text n greater or equal than 0,
    We are asked to find the expansion of left parenthesis 4 straight g plus 2 straight h right parenthesis to the power of 4  using Binomial theorem.
    Step 1 of 2:
    The given expression isleft parenthesis 4 straight g plus 2 straight h right parenthesis to the power of 4. Here, the value of n = 4, so we would have 4+1 = 5 terms in the expansion. The values arestraight x equals 4 straight g straight & straight y equals 2 straight h.
    Step 2 of 2:
    Substitute the values in the binomial equation to get the terms of the expansion; table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell left parenthesis 4 g plus 2 h right parenthesis to the power of 4 equals 4 C subscript 0 left parenthesis 4 g right parenthesis to the power of 4 plus 4 C subscript 1 left parenthesis 4 g right parenthesis cubed left parenthesis 2 h right parenthesis plus 4 C subscript 2 left parenthesis 4 g right parenthesis squared left parenthesis 2 h right parenthesis squared plus 4 C subscript 3 left parenthesis 4 g right parenthesis left parenthesis 2 h right parenthesis cubed plus 4 C subscript 4 left parenthesis 2 h right parenthesis to the power of 4 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 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 exponent end cell row cell space space equals 256 g to the power of 4 plus 4 open parentheses 64 g cubed close parentheses left parenthesis 2 h right parenthesis plus 6 open parentheses 16 g squared close parentheses open parentheses 4 h squared close parentheses plus 4 left parenthesis 4 g right parenthesis open parentheses 8 h cubed close parentheses plus 16 h to the power of 4 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 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 256 g to the power of 4 plus 512 g cubed h plus 384 g squared h squared plus 128 g h cubed plus 16 h to the power of 4 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 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 space space space space space space end cell end table
    Thus, the expansion of the expression is left parenthesis 4 straight g plus 2 straight h right parenthesis to the power of 4 equals 256 straight g to the power of 4 plus 512 straight g cubed straight h plus 384 straight g squared straight h squared plus 128 gh cubed plus 16 straight h to the power of 4
    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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