Math

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How many terms are in the expansion of (a+b+c)^3 after like terms have been combined?

1. I don't know what this question is asking
2 and I don't know how to start it

  • Math -

    (a+b+c)^3
    =(a+b+c)(a+b+c)(a+b+c)
    = (a+b+c)(a^2 + b^2 + c^2 + 2ab + 2bc + 2ac) , I multiplied and simplified the last two factors.
    = .....
    See if you finish the expansion, add up all like terms and then count the number of terms.

  • Math -

    Take (a+b)².
    After expansion, it becomes:
    a²+ab+ba+b²
    There are physically 4 terms. But if we combine like terms,
    ab+ba=2ab
    So we end up with three terms:
    a²+2ab+b².

    The question would like an answer for the case of (a+b+c)³.

    The expression is homogeneous, meaning that the sum of the exponents of every term is three.

    So now we can list the ways a, b and c can be multiplied together with a sum of exponent of three, and then count the number of different terms possible.

    Here's the list:

    a²b
    a²c
    ab²
    abc
    ac²

    b²c
    bc²

  • Math -

    Term? Meaning each number and/or variable separated by a + or - sign. example 2x^2+4x-3 has three terms?

  • Math -

    Exactly!
    For example,
    -3x²
    is a term,
    -3 is the coefficient
    x is the variable,and
    ² is the exponent.
    The sign is part of the coefficient.

  • Math -

    this is crzy thing

  • Math -

    "Like terms" are any terms in the multiplied-out product that have the same powers of a, b or c.

    (a + b + c)^2 = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

    (a + b + c)^3 = (a + b + c)( a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
    = (a^3 + ab^2 + ac^2 + 2a2b + 2 abc +2a^2c) + (a^2b + b^3 + bc^2 + 2ab^2 + 2abc + 2 b^2c) + (a^2c + b^2c + c^3 + 2abc + 2ac^2 + 2bc^2)
    = a^3 + b^3 + c^3 + 6abc +3ab^2 + 3ac^2 + 3bc^2 +3a^2b + 3b^2c +3a^2c

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