aly asked in Science & MathematicsMathematics · 1 decade ago

do you see a pattern?

(a+b)^8 = a^8 + 8 a^7 b + 28 a^6 b^2 + 56 a^5 b^3 + 70 a^4 b^4 + 56 a^3 b^5 + 28 a^2 b^6 + 8 a b^7 + b^8

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  • 1 decade ago
    Favorite Answer

    Yes, and this is because of Binomial theorem.

    According to Binomial theorem

    (a+b)^n = sum(k=0 to k=n)[(a^k) * (b^(n-k))]

    So, with each term in your expantion, we observe 3 things

    1. Power of one of the variable increases from zero to n.

    2. Power of remaining of the variable decreases from n to zero.

    3. The coefficient of each term from starting to last will have coefficient (n perm k), where n is the power of (a+b) on RHS and k is the number of the term you are evaluating.

    I hope this answers your question.

    All the best.

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  • Broden
    Lv 4
    1 decade ago

    . . . . . . k=8

    (a+b)^8=∑ C(8,k)*(a^(8-k)*b^(k)

    . . . . . . k=0

    And C(8,k)=8!/[k!*(8-k)!]

    C(8,k) is called the binominial Coefficient

    0!≡ 1 (Definition)

    8!=8*7*6*5*4*3*2*1

    k!= k* (k-1)*......1

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  • Anonymous
    1 decade ago

    Yes, two patterns

    Exponent for a decreases and exponent for b increases.

    Also coefficient increases by 14 the decreases by 14

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  • 1 decade ago

    Yes, I do. It's called the 'binomial theorem' and you really should learn all about it because it's fairly basic to a lot of mathematics.

    $5 says that if you type 'binomial theorem' into a search engine, you'll get a couple hundred thousand(or more) hits.

    Doug

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  • kevin!
    Lv 5
    1 decade ago

    Exponent of a decreases...

    Exponent of b increases...

    The coefficients are the same as the 9th row of Pascal's triangle.

    In general,

    (a + b)^n

    The coefficients are the same as the (n + 1)th row of Pascal's triangle.

    ^_^

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