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Question

Use the binomial theorem to expand the expressions: open parentheses x cubed plus y squared close parentheses to the power of 6

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 x cubed plus y squared close parentheses to the power of 6 using Binomial theorem.

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


     Step 1 of 2:
    The given expression is open parentheses x cubed plus y squared close parentheses to the power of 6 . Here, n=6, so we would have 6+1=7 terms in the expansion. The values of x equals x cubed straight & y equals y squared
    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 x cubed plus y squared close parentheses to the power of 6 equals 6 C subscript 0 open parentheses x cubed close parentheses to the power of 6 plus 6 C subscript 1 open parentheses x cubed close parentheses to the power of 5 open parentheses y squared close parentheses plus 6 C subscript 2 open parentheses x cubed close parentheses to the power of 4 open parentheses y squared close parentheses squared plus 6 C subscript 3 open parentheses x cubed close parentheses cubed open parentheses y squared close parentheses cubed end cell row cell plus 6 C subscript 4 open parentheses x cubed close parentheses squared open parentheses y squared close parentheses to the power of 4 plus 6 C subscript 5 open parentheses x cubed close parentheses open parentheses y squared close parentheses to the power of 5 plus 6 C subscript 6 open parentheses y squared close parentheses to the power of 6 end cell row cell equals x to the power of 18 plus 6 open parentheses x to the power of 15 close parentheses y squared plus 15 open parentheses x to the power of 12 close parentheses open parentheses y to the power of 4 close parentheses plus 20 open parentheses x to the power of 9 close parentheses open parentheses y to the power of 6 close parentheses plus 15 open parentheses x to the power of 6 close parentheses open parentheses y to the power of 8 close parentheses plus 6 open parentheses x cubed close parentheses open parentheses y to the power of 10 close parentheses plus y to the power of 12 end cell row cell equals x to the power of 18 plus 6 x to the power of 15 y squared plus 15 x to the power of 12 y to the power of 4 plus 20 x to the power of 9 y to the power of 6 plus 15 x to the power of 6 y to the power of 8 plus 6 x cubed y to the power of 10 plus y to the power of 12 end cell end table
    Thus, the expansion of the expression is:  open parentheses x cubed plus y squared close parentheses to the power of 6 equals x to the power of 18 plus 6 x to the power of 15 y squared plus 15 x to the power of 12 y to the power of 4 plus 20 x to the power of 9 y to the power of 6 plus 15 x to the power of 6 y to the power of 8 plus 6 x cubed y to the power of 10 plus y to the power of 12

    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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