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Question

Use binomial theorem to expand (x - 1)7

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 , here n ≥ 0 . We are asked to use binomial theorem to expand (x - 1)7 .

The correct answer is: (x + y)n



    Step 1 of 2:
    The given expression is (x - 1)7  , here x = x & y = -1 . The value of n=7, hence there are 7+1=8 terms in the expressions.
    Step 2 of 2:
    Substitute the values of (x - 1)7 in the binomial equation to get 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 x minus 1 right parenthesis to the power of 7 equals 7 C subscript 0 left parenthesis x right parenthesis to the power of 7 plus 7 C subscript 1 left parenthesis x right parenthesis to the power of 6 left parenthesis negative 1 right parenthesis plus 7 C subscript 2 left parenthesis x right parenthesis to the power of 5 left parenthesis negative 1 right parenthesis squared plus 7 C subscript 3 left parenthesis x right parenthesis to the power of 4 left parenthesis negative 1 right parenthesis cubed plus 7 C subscript 4 left parenthesis x right parenthesis cubed left parenthesis negative 1 right parenthesis to the power of 4 plus 7 C subscript 5 left parenthesis x right parenthesis squared left parenthesis negative 1 right parenthesis to the power of 5 plus 7 C subscript 6 left parenthesis x right parenthesis left parenthesis negative 1 right parenthesis to the power of 6 plus 7 C subscript 7 left parenthesis negative 1 right parenthesis to the power of 7 end cell row cell equals x to the power of 7 plus 7 x to the power of 6 left parenthesis negative 1 right parenthesis plus 21 x to the power of 5 left parenthesis 1 right parenthesis plus 35 x to the power of 4 left parenthesis negative 1 right parenthesis plus 35 x cubed left parenthesis 1 right parenthesis plus 21 x squared left parenthesis negative 1 right parenthesis plus 7 x left parenthesis 1 right parenthesis plus left parenthesis negative 1 right parenthesis end cell row cell equals x to the power of straight ⊤ minus 7 x to the power of 6 plus 21 x to the power of 5 minus 35 x to the power of 4 plus 35 x cubed minus 21 x squared plus 7 x minus 1 end cell end table
    Thus, the expansion is: left parenthesis x minus 1 right parenthesis to the power of 7 equals x to the power of 7 minus 7 x to the power of 6 plus 21 x to the power of 5 minus 35 x to the power of 4 plus 35 x cubed minus 21 x squared plus 7 x minus 1

    For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.

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