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Question

Use the binomial theorem to expand the expressions: open parentheses 2 x plus 1 third close parentheses cubed

hintHint:

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 open parentheses 2 x plus 1 third close parentheses cubed using Binomial theorem.

The correct answer is: (n+1)th row of the triangle.


    Step 1 of 2:
    The given expression is open parentheses 2 x plus 1 third close parentheses cubed . Here, n=3, so we would have 3+1=4 terms in the expansion of the expression. The value of x equals 2 x straight & y equals 1 third .
    Step 2 of 2:
    Substitute the values in the binomial equation to get the expansion of the expression;

    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 open parentheses 2 x plus 1 third close parentheses cubed equals 3 C subscript 0 left parenthesis 2 x right parenthesis cubed plus 3 C subscript 1 left parenthesis 2 x right parenthesis squared open parentheses 1 third close parentheses plus 3 C subscript 2 left parenthesis 2 x right parenthesis open parentheses 1 third close parentheses squared plus 3 C subscript 3 open parentheses 1 third close parentheses cubed end cell row cell equals 8 x cubed plus 3 open parentheses 4 x squared close parentheses open parentheses 1 third close parentheses plus 3 left parenthesis 2 x right parenthesis open parentheses 1 over 9 close parentheses plus 1 over 27 end cell row cell equals 8 x cubed plus 4 x squared plus fraction numerator 2 x over denominator 3 end fraction plus 1 over 27 end cell end table
    Thus, the expansion is: open parentheses 2 x plus 1 third close parentheses cubed equals 8 x cubed plus 4 x squared plus fraction numerator 2 x over denominator 3 end fraction plus 1 over 27.

    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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