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Math Help - binomial theorem?

  1. #1
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    binomial theorem?

    Use pascals triangle to expand each binomial

    (square root c-1)^6

    Can anyone help me expand the binomial I know the row I am supposed to use

    1 6 15 20 15 6 1

    But the square root is slightly confusing me?
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  2. #2
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    Quote Originally Posted by homeylova223 View Post
    Use pascals triangle to expand each binomial

    (square root c-1)^6

    Can anyone help me expand the binomial I know the row I am supposed to use

    1 6 15 20 15 6 1

    But the square root is slightly confusing me?
    You have
    (a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + ~...

    and a = \sqrt{c} and b = -1

    Thus
    (\sqrt{c} - 1)^6 = (\sqrt{c})^6 + 6(\sqrt{c})^5(-1) + 15(\sqrt{c})^4(-1)^2 + ~...

    -Dan
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    like topsquark shows.

    you just have to remember the structure. put that down and plug in your values and coefficients. then do the algebra for each term.

    so think, the triangle starts with row 0 and you are at the 6th power so the 7th row and therefore 7 terms. or think 6+1 terms so 7 terms in total for that row.

    the terms are homogeneous which means the sum of all the exponents in each term is the same number. 6 in your case. just shift one power each term form one exponent to the other. a exponent descends as b exponent assents.

    now make your structure



    then just plug in your values and do the algebra.

    you can calculate structure in your head very easily. say you want 11th term of 20th power. 20th power=row 21, so 21-11=10 a=10, 10+10=20 b=10. notice that because the number of terms 21 is odd the middle term has identicle coefients of a and b. now use binomial theorem to calculate the coeffient.
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  4. #4
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    Quote Originally Posted by skoker View Post
    like topsquark shows.

    you just have to remember the structure. put that down and plug in your values and coefficients. then do the algebra for each term.

    so think, the triangle starts with row 0 and you are at the 6th power so the 7th row and therefore 7 terms. or think 6+1 terms so 7 terms in total for that row.

    the terms are homogeneous which means the sum of all the exponents in each term is the same number. 6 in your case. just shift one power each term form one exponent to the other. a exponent descends as b exponent assents.

    now make your structure



    then just plug in your values and do the algebra.

    you can calculate structure in your head very easily. say you want 11th term of 20th power. 20th power=row 21, so 21-11=10 a=10, 10+10=20 b=10. notice that because the number of terms 21 is odd the middle term has identicle coefients of a and b. now use binomial theorem to calculate the coeffient.
    Don't forget the Pascal's Triangle terms in the "structure".
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