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Question

Use binomial theorem to expand expressionleft parenthesis x plus y right parenthesis to the power of 7.

The correct answer is: Thus, the expansion is:


    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 , here n greater or equal than 0 .
    We are asked to find the binomial expansion of left parenthesis x plus y right parenthesis to the power of 7 using binomial theorem.
    Step 1 of 2:
    The given expression is left parenthesis x plus y right parenthesis to the power of 7 . Here, the value of n=7. Thus, there are 7+1=8 terms in the expansion.
    Step 2 of 2:
    Substitute the values in the expansion to get the terms of the expansion. Thus, we have:
    left parenthesis x plus y right parenthesis to the power of 7 equals 7 C subscript 0 x to the power of 7 plus 7 C subscript 1 x to the power of 6 left parenthesis y right parenthesis plus 7 C subscript 2 left parenthesis x right parenthesis to the power of 5 left parenthesis y right parenthesis squared plus 7 C subscript 3 left parenthesis x right parenthesis to the power of 4 left parenthesis y right parenthesis cubed plus 7 C subscript 4 left parenthesis x right parenthesis cubed left parenthesis y right parenthesis to the power of 4 plus 7 C subscript 5 left parenthesis x right parenthesis squared left parenthesis y right parenthesis to the power of 5 plus 7 C subscript 6 left parenthesis x right parenthesis left parenthesis y right parenthesis to the power of 6 plus 7 C subscript 7 left parenthesis y right parenthesis to the power of 7
    equals x to the power of 7 plus 7 x to the power of 6 y plus 21 x to the power of 5 y squared plus 35 x to the power of 4 y cubed plus 35 x cubed y to the power of 4 plus 21 x squared y to the power of 5 plus 7 x y to the power of 6 plus y to the power of 7
    Thus, the expansion is:
    left parenthesis x plus y right parenthesis to the power of 7 equals x to the power of 7 plus 7 x to the power of 6 y plus 21 x to the power of 5 y squared plus 35 x to the power of 4 y cubed plus 35 x cubed y to the power of 4 plus 21 x squared y to the power of 5 plus 7 x y to the power of 6 plus y to the power of 7
    .Note:
    The answer can be found using the Pascal’s triangle. For an expression left parenthesis x plus y right parenthesis to the power of n , we would have n+ 1 term.

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