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Question

Use the binomial theorem to expand the expressions: left parenthesis 2 a minus b right parenthesis to the power of 5

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 find the expansion of left parenthesis 2 a minus b right parenthesis to the power of 5 using Binomial theorem.

The correct answer is: triangle


     Step 1 of 2:
    The given expression is left parenthesis 2 a minus b right parenthesis to the power of 5 where n=5. So, we would have 5+1=6 terms in the expansion. The values of x equals 2 a straight & y equals negative b .
    Step 2 of 2:
    Substitute the values in the binomial equation to get the answer:

    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 2 a minus b right parenthesis to the power of 5 equals 5 C subscript 0 left parenthesis 2 a right parenthesis to the power of 5 plus 5 C subscript 1 left parenthesis 2 a right parenthesis to the power of 4 left parenthesis negative b right parenthesis plus 5 C subscript 2 left parenthesis 2 a right parenthesis cubed left parenthesis negative b right parenthesis squared plus 5 C subscript 3 left parenthesis 2 a right parenthesis squared left parenthesis negative b right parenthesis cubed plus 5 C subscript 4 left parenthesis 2 a right parenthesis left parenthesis negative b right parenthesis to the power of 4 plus 5 C subscript 5 left parenthesis negative b right parenthesis to the power of 5 end cell row cell equals 32 a to the power of 5 plus 5 open parentheses 16 a to the power of 4 close parentheses left parenthesis negative b right parenthesis plus 10 open parentheses 8 a cubed close parentheses open parentheses b squared close parentheses plus 10 open parentheses 4 a squared close parentheses open parentheses negative b cubed close parentheses plus 5 left parenthesis 2 a right parenthesis open parentheses b to the power of 4 close parentheses plus left parenthesis negative b right parenthesis to the power of 5 end cell row cell equals 32 a to the power of 5 minus 80 a to the power of 4 b plus 80 a cubed b squared minus 40 a squared b cubed plus 10 a b to the power of 4 minus b to the power of 5 end cell end table
    Thus, the expansion is ; left parenthesis 2 a minus b right parenthesis to the power of 5 equals 32 a to the power of 5 minus 80 a to the power of 4 b plus 80 a cubed b squared minus 40 a squared b cubed plus 10 a b to the power of 4 minus b to the power of 5

    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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