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29 May, 18:42

How to find the fourth term of a binomial expansion?

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  1. 29 May, 19:23
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    So let's look at how to expand a binomial ...

    Start with just the variable exponents which will be numbered in ascending and descending order.

    example (without coefficients)

    (x+y) ^3 = x^3 (y^0) + x^2 (y^1) + x^1 (y^2) + x^0 (y^3)

    If the binomial is (x-y) ^3 the + / - signs will toggle like so ... (again leaving out coefficients)

    (x-y) ^3 = x^3 (y^0) - x^2 (y^1) + x^1 (y^2) - x^0 (y^3)

    If there is a coefficient or power inside the binomial like (x^2+2y) ^3 these are also raised to the appropriate power.

    (x^2+2y) ^3 = (x^2) ^3 (y^0) (2^0) + (x^2) ^2 (y^1) (2^1) + ...

    now there are also coefficients for each term which follow a combination pattern of Pascal triangle. C (3,0); C (3,1); C (3,2); C (3,3)

    these get multiplied by each term.

    So the 4th term of (x^2-2y) ^3

    will be negative

    have coefficient of: C (3,3) * (2^3) = 8

    variables: x^0 (y^3) = y^3

    = - 8y^3
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