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Question

Use binomial theorem to expand expression left parenthesis d minus 1 right parenthesis to the power of 4

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 d minus 1 right parenthesis to the power of 4 .
    Step 1 of 2:
    The given expression is left parenthesis d minus 1 right parenthesis to the power of 4 . Here x = d   straight & y equals negative 1and value of n=4. Thus, the number of terms in
    the expansion of left parenthesis d minus 1 right parenthesis to the power of 4 is 5.
    Substitute the values in the binomial expansion to get the required terms. Thus, we have:
    left parenthesis d minus 1 right parenthesis to the power of 4 equals 4 C subscript 0 left parenthesis d right parenthesis to the power of 4 plus 4 C subscript 1 left parenthesis d right parenthesis cubed left parenthesis negative 1 right parenthesis plus 4 C subscript 2 left parenthesis d right parenthesis squared left parenthesis negative 1 right parenthesis squared plus 4 C subscript 3 left parenthesis d right parenthesis left parenthesis negative 1 right parenthesis cubed plus 4 C subscript 4 left parenthesis negative 1 right parenthesis to the power of 4
    equals d to the power of 4 plus 4 d cubed left parenthesis negative 1 right parenthesis plus 6 d squared left parenthesis negative 1 right parenthesis squared plus 4 d left parenthesis negative 1 right parenthesis cubed plus left parenthesis negative 1 right parenthesis to the power of 4
    equals d to the power of 4 minus 4 d cubed plus 6 d squared minus 4 d plus 1
    Thus, the expansion is: .left parenthesis d minus 1 right parenthesis to the power of 4 equals d to the power of 4 minus 4 d cubed plus 6 d squared minus 4 d plus 1
    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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