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Math Help - GCSE Algebra - Simplification.

  1. #1
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    GCSE Algebra - Simplification.

    Hello - I've been trying this problem today and it's very frustrating, I've spent hours on it and don't feel like I've got anywhere...

    Here's a link to the question and the answer (book)

    So - I see that the problem is essentially a sum of cubes.... but I just can't seem to do it... I'm sure that my method must be wrong, as it's very convoluted. I've done problems that have had simpler sum of cubes, but not like this.

    Anyway, my working is in two parts -

    Here's the first one

    heres the second.

    Really, I've spent hours on this (embarrassingly)... Obviously this isn't the only paper I've consumed in the process but it's the most recent...

    thanks
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  2. #2
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    Re: GCSE Algebra - Simplification.

    First, thank you for showing work.

    Second, at least in your first go at it, you use a and b in two senses, which is likely to scramble your brain.

    General factoring: $(x + y)(x^2 - xy + y^2) = x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3 = x^3 + y^3.$

    So let x = (a - 2b) and y = (2a - b).

    $\therefore (a - 2b)^3 + (2a - b)^3 = x^3 + y^3 = (x + y)(x^2 - xy + y^2) =$

    $\{(a - 2b) + (2a - b)\}\{(a - 2b)^2 - (a - 2b)(2a - b) + (2a - b)^2\} =$

    $(3a - 3b)\{a^2 - 4ab + 4b^2 - (2a^2 - 5ab + 2b^2) + 4a^2 - 4ab + b^2\} =$

    $3(a - b)(3a^2 - 3ab + 3b^2) = 3 * 3(a - b)(a^2 - ab + b^2) = 9(a - b)(a^2 - ab + b^2).$
    Thanks from topsquark and silverpen
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  3. #3
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    Re: GCSE Algebra - Simplification.

    First, thank you for showing work
    No worries

    Second, at least in your first go at it,
    Ah, they were both the same go - it went over the edge of the page


    likely to scramble your brain
    Yes - whilst working through this problem I was sure that there must be a better / more pragmatic method to use, but I couldn't think of one... Honestly, this problem has ruined most of the day so thanks for the help!!

    I actually did a bit more along the lines of what you posted - here's a link to that

    I have to go out now unfortunately - but I look forward to properly looking through the response and comparing when I get back...

    Thanks a lot, this problem completely beat me!
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    Re: GCSE Algebra - Simplification.

    Quote Originally Posted by silverpen View Post
    Yes - whilst working through this problem I was sure that there must be a better / more pragmatic method to use, but I couldn't think of one... Honestly, this problem has ruined most of the day so thanks for the help!!

    I actually did a bit more along the lines of what you posted - here's a link to that

    Thanks a lot, this problem completely beat me!
    When you want to apply a general formula to a specific problem, it will reduce possible confusion if you express the general formula using letters completely different from those being used in the problem. As for the url shown above, you were on exactly the same track as I was. You had only a step or two more to go.
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    Re: GCSE Algebra - Simplification.

    Thanks Jeff - OK I've had chance to have a look at this now and to be honest I'm still not 100%!

    Is this how you would multiply out this problem? LINK TO IMAGE

    On of the things that I'm unsure about is the double parentheses in the second bracket set ( where -xy would be in the simpler example)

    I'm also unsure how you went from the second line to the third line in the example you gave - is that just multiplying out or was there simplification in that as well?

    I'm going to carry on working on it, just thought I'd post that.
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  6. #6
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    Re: GCSE Algebra - Simplification.

    Quote Originally Posted by silverpen View Post
    Thanks Jeff - OK I've had chance to have a look at this now and to be honest I'm still not 100%!

    Is this how you would multiply out this problem? LINK TO IMAGE

    On of the things that I'm unsure about is the double parentheses in the second bracket set ( where -xy would be in the simpler example)

    I'm also unsure how you went from the second line to the third line in the example you gave - is that just multiplying out or was there simplification in that as well?

    I'm going to carry on working on it, just thought I'd post that.
    Let's start with your image. There is an error in it. You indicate that when you multiply a binomial (two expressions joined by + or -) times a trinomial (three expressions joined by + or -) that represents 8 multiplications. This is just wrong: it represents 6 multiplications. You can see that it is 6 by looking at the general derivation using x and y in my previous post.

    However, one reason that you study simplification is to eliminate even thinking about multiplication of really complex expressions if you can. I am not sure what lines you are specifying, but I suspect it is the transition from two expressions in curly braces to one expression in parentheses and the other still in curly braces. In the first expression, I simplified. In the second, I multiplied out. Of course then I simplified again. Only after I had simplified within the individual parentheses would I even contemplate doing any other operation because I am lazy and error prone.
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  7. #7
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    Re: GCSE Algebra - Simplification.

    Let's start with your image. There is an error in it. You indicate that when you multiply a binomial (two expressions joined by + or -) times a trinomial (three expressions joined by + or -) that represents 8 multiplications. This is just wrong: it represents 6 multiplications. You can see that it is 6 by looking at the general derivation using x and y in my previous post.
    Yes - it would be best to simplify the two parentheses first then

    Maybe it's best I show my working again - here's a link to some new working.

    What I can't see is how you go about the steps of simplification that you have done... I am consistently failing and maybe that's just me or maybe there's an easier way to go about it IDK
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    Re: GCSE Algebra - Simplification.

    ah no - I see the errors on that hang on!
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    Re: GCSE Algebra - Simplification.

    Quote Originally Posted by silverpen View Post
    Yes - it would be best to simplify the two parentheses first then

    Maybe it's best I show my working again - here's a link to some new working.

    What I can't see is how you go about the steps of simplification that you have done... I am consistently failing and maybe that's just me or maybe there's an easier way to go about it IDK
    I presume you saw the error you made in doing the squares.

    Here is what I think: you have got so frustrated with this problem that you are making careless errors. I think you can really do this now if you just take care: you are almost there.
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    Re: GCSE Algebra - Simplification.

    you have got so frustrated with this problem that you are making careless errors
    Yeah I think so to, I just went through the problem again here

    And it wasn't too bad - I'm going to have a look at a few others in the same problem set, thanks for you help Jeff!


    EDIT - I'm not sure If I can mark this thread solved? I looked for the option and couldn't find it.
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  11. #11
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    Re: GCSE Algebra - Simplification.

    You're welcome. And congratulations on finally plowing through it. Don't worry about marking the problem solved: other students may find it helpful in the future.
    Thanks from silverpen
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  12. #12
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    Re: GCSE Algebra - Simplification.

    Don't worry about marking the problem solved: other students may find it helpful in the future
    OK cool - some places like it marked so other's either know that OP doesn't require any further help and so that future users know there's an answer at the end of it

    Cheers though Jeff!
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