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    How do I expand and simplify these brackets using pascal's triangle?

    (1 + x + x^2)^3

    Thanks.
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    It's actually the binomial expansion you need to use and you know (1+y)^n = 1 + nC1.y + nC2.y^2 + nC3.y^3 + ..

    then let y = x+x^2

    another way;
    (1+x+x^2)^3
    = ((x+1/2)^2+3/4)^3 (by completing the square)

    Let y = x+1/2
    = (y^2 + 3/4)^3

    should be easy to expand and replace y for x
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    dont ever use pascal triangle for this - use binomial.

    Pascals triangle in my eyes should only ever be used as some fun for intrested 11 year olds
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    It's a trinomial

    http://en.wikipedia.org/wiki/Trinomial_expansion

    (Original post by eulerwaswrong)

    Pascals triangle in my eyes should only ever be used as some fun for intrested 11 year olds
    I really think you're underestimating Pascal's triangle there.
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    (Original post by rnd)
    It's a trinomial

    http://en.wikipedia.org/wiki/Trinomial_expansion



    I really think you're underestimating Pascal's triangle there.
    yea you're right - i was just trying to say that you should never use it for binomial expansions - you should always use binomial thm
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    (Original post by eulerwaswrong)
    yea you're right - i was just trying to say that you should never use it for binomial expansions - you should always use binomial thm
    Why?
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    (Original post by DFranklin)
    Why?
    because its not maths like - remembering a series of numbers just doesnt seem the "proper" way to do it.

    And i never get the reason why maths teachers tell their pupils to do it using pascals in c1, then tell them to use binomial in c4. In the end the pupils will need to know how to use binomial - so why not do it with them in c1 - it makes more sense, do it the proper way first time instead of faffing about with some silly rule.
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    (Original post by rnd)
    I really think you're underestimating Pascal's triangle there.


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    (Original post by eulerwaswrong)
    because its not maths like - remembering a series of numbers just doesnt seem the "proper" way to do it.
    But Pascal's triangle isn't a sequence to be remembered, any more than \binom{n}{r} (r = 0, 1, ... , n) is.

    Moreover, if you consider how you calculate (1+x)^n from (1+x)^{n-1}, you see that the process is closely analogous to how you calculate each row of Pascal's triangle from the next. It is very easy to explain why the coefficients in (1+x)^n match the values in the n+1 th row of Pascal's triangle.

    In contrast, it's really quite difficult to explain why the formula (1+x)^n = \sum_0^n \frac{x^k n!}{k!(n-k)!} works. So if anything is a something "just being remembered without understanding" , I'd say it's the latter.

    And i never get the reason why maths teachers tell their pupils to do it using pascals in c1, then tell them to use binomial in c4. In the end the pupils will need to know how to use binomial - so why not do it with them in c1 - it makes more sense, do it the proper way first time instead of faffing about with some silly rule.
    In c4, you will need to deal with (1+x)^n when n isn't an integer, at which point you need the Binomial Theorem.

    There's a very real sense in which the (full) Binomial Theorem is a generalisation of Pascal's Triangle. But Pascal's Triangle is a far more fundamental result, and it makes a lot of sense for it to be taught first.
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    (Original post by DFranklin)
    But Pascal's triangle isn't a sequence to be remembered, any more than \binom{n}{r} (r = 0, 1, ... , n) is.

    Moreover, if you consider how you calculate (1+x)^n from (1+x)^{n-1}, you see that the process is closely analogous to how you calculate each row of Pascal's triangle from the next. It is very easy to explain why the coefficients in (1+x)^n match the values in the n+1 th row of Pascal's triangle.

    In contrast, it's really quite difficult to explain why the formula (1+x)^n = \sum_0^n \frac{x^k n!}{k!(n-k)!} works. So if anything is a something "just being remembered without understanding" , I'd say it's the latter.

    In c4, you will need to deal with (1+x)^n when n isn't an integer, at which point you need the Binomial Theorem.

    There's a very real sense in which the (full) Binomial Theorem is a generalisation of Pascal's Triangle. But Pascal's Triangle is a far more fundamental result, and it makes a lot of sense for it to be taught first.
    well your leagues above me in mathematical ability and i cant believe im actually arguing against you on something remotely mathematical but..........................

    What you say is true although people dont get "taught" pascals triangle the way you are thinking about it. People get taught to remember the numbers and nothing more. They dont "understand" pascals triangle - they only know the rows of numbers.

    At least with the binomial theorem you might not know where it fundamentally comes from but at least you arent just told to remember sets of numbers, it applies to more situations, its more adaptable. Its just better - you might not know were it comes from exactly - but it can be explained relatively easily and it just works for all. There is no real need to be ever shown pascals triangle for expanding brackets.


    understanding pascals triangle > using binomial theorem > remembering pascals triangle
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    (Original post by eulerwaswrong)
    What you say is true although people dont get "taught" pascals triangle the way you are thinking about it. People get taught to remember the numbers and nothing more. They dont "understand" pascals triangle - they only know the rows of numbers.
    Various things:

    (a) Then you're arguing about how they're taught, which doesn't change the fact that Pascal's triangle is still a perfectly valid way of expanding brackets.
    (b) You are in a better position to know than me, but I'm very surprised to hear people are taught Pascal's triangle without knowing the rule for going from one row to the next. If you google "Pascal's Triangle', pretty much every link starts by explaining exactly that. In contrast, very few accounts of the binomial theorem (at A-level) will explain why it works.
    (c) To be honest, if you're talking about (1+x)^n for n less than about 5, you should know the relevant rows of Pascal's triangle. It's far superior to sitting there working out 4! / 2! 2!.

    At least with the binomial theorem you might not know where it fundamentally comes from but at least you arent just told to remember sets of numbers, it applies to more situations, its more adaptable. Its just better - you might not know were it comes from exactly - but it can be explained relatively easily and it just works for all.
    There's a huge difference in comprehensibility between Pascal's triangle (which can be taught at GCSE level - it was in my day), and the Binomial Theorem for arbitrary exponents. And it doesn't work for all - the resultant series diverges if |x| > 1 unless it terminates.

    There is no real need to be ever shown pascals triangle for expanding brackets.
    Except that it's much much easier to understand.

    To give a different example: What you are taught about integration at A-level is largely complete nonsense. You will throw nearly all of it away and start again when you do "proper" Riemann Integration. And to cap it all, you will then throw nearly all of that away and start again when you do Lebesgue Integration. And yet you won't find people arguing that we should be teaching Lebesgue Integration at A-level. Because A-level integration is generally "good enough", and it gets you comfortable with the concepts you will later need for Lebesgue Integration. It also lets you solve the actual problems you need to solve.

    Similarly, for many things, Pascal's Triangle is good enough (it is, after all, exactly as good as the Binomial Theorem in the finite case). It's easy to understand why it works, and it lets you do the expansions that you need to do.
 
 
 
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