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Math Help - real quick question about factoring

  1. #1
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    real quick question about factoring

    I have this question in my book that reads:

    ax - bx + by - ay

    it shows me the entire process, even the answer but one part has me confused and is not explained.

    x(a-b)+ y(-1)(a-b)
    next to it, it says "Factoring out -1 to reverse b-a"

    Im just really confused as to where they got that -1 from. It just seems like they pulled it out of thin air.

    --------------------------

    heres the entire process up until that point:

    ax - bx + by - ay = (ax - bx) + (by-ay)
    =x(a-b)+y(b-a)
    =x(a-b)+ y(-1)(a-b)
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  2. #2
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    If you have the expression b - a you can factor out -1 by dividing both terms by -1, the terms being b and -a. \frac{b}{-1} = -b, and \frac{-a}{-1} = a, so you end up with -1(-b + a), or -1(a - b). Make sense? Basically you can pull out whatever number you want, as long as you divide accordingly. I could factor 543 out of b - a, and I would end up with 543 (\frac{1}{543}b - \frac{1}{543}a)--multiply that out if you don't believe that it's the same thing as b - a. In this case they're pulling out a -1.
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  3. #3
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    b-a = -(a-b) = -1 \cdot (a-b)
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by NecroWinter View Post
    I have this question in my book that reads:

    ax - bx + by - ay

    it shows me the entire process, even the answer but one part has me confused and is not explained.

    x(a-b)+ y(-1)(a-b)
    next to it, it says "Factoring out -1 to reverse b-a"

    Im just really confused as to where they got that -1 from. It just seems like they pulled it out of thin air.

    --------------------------

    heres the entire process up until that point:

    ax - bx + by - ay = (ax - bx) + (by-ay)
    =x(a-b)+y(b-a)
    =x(a-b)+ y(-1)(a-b)
    several ways you can think about this. Here's one:

    when you factor a term from an expression, you actually divide each term in the expression by the factored-out term in order to factor it out. for example:

    Say you wanted to factor \displaystyle x^3 + x. Of course you would factor out the common \displaystyle x to get \displaystyle x(x^2 + 1)

    But where did the \displaystyle x^2 and the \displaystyle +1 come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an \displaystyle x, so divide the \displaystyle x^3 by \displaystyle x, and that's where your \displaystyle x^2 comes from. Your 1 comes from dividing \displaystyle x by \displaystyle x.

    So in other words, what you did was \displaystyle x \left( \frac {x^3}x + \frac xx \right) = x(x^2 + 1)

    (you can also think of this as multiplying and dividing by x at the same time. this way, you're multiplying by 1 and so you're not changing the value of your expression, just how it looks)

    Similarly, \displaystyle b - a = -1 \left( \frac b{-1} - \frac a{-1} \right) = -(a - b).

    We usually don't think of factoring in this complicated way, but that's what we're doing when we factor something. as for factoring out -1, you just think of it as factoring out a negative sign, when you do this, you change all the signs of the terms you factored the -1 from. so b - a = -(a - b), the +b became -b and the -a became +a when you factor out a sign. If you multiply out, you get the original expression, which is a good way to check yourself.

    Capice? I don't know. I feel like I'm being way too complicated with everything today... maybe I shouldn't be explaining stuff to people
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    several ways you can think about this. Here's one:

    when you factor a term from an expression, you actually divide each term in the expression by the factored-out term in order to factor it out. for example:

    Say you wanted to factor \displaystyle x^3 + x. Of course you would factor out the common \displaystyle x to get \displaystyle x(x^2 + 1)

    But where did the \displaystyle x^2 and the \displaystyle +1 come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an \displaystyle x, so divide the \displaystyle x^3 by \displaystyle x, and that's where your \displaystyle x^2 comes from. Your 1 comes from dividing \displaystyle x by \displaystyle x.

    So in other words, what you did was \displaystyle x \left( \frac {x^3}x + \frac xx \right) = x(x^2 + 1)

    (you can also think of this as multiplying and dividing by x at the same time. this way, you're multiplying by 1 and so you're not changing the value of your expression, just how it looks)

    Similarly, \displaystyle b - a = -1 \left( \frac b{-1} - \frac a{-1} \right) = -(a - b).

    We usually don't think of factoring in this complicated way, but that's what we're doing when we factor something. as for factoring out -1, you just think of it as factoring out a negative sign, when you do this, you change all the signs of the terms you factored the -1 from. so b - a = -(a - b), the +b became -b and the -a became +a when you factor out a sign. If you multiply out, you get the original expression, which is a good way to check yourself.

    Capice? I don't know. I feel like I'm being way too complicated with everything today... maybe I shouldn't be explaining stuff to people
    No, that was actually a fantastic explaination. You dont give yourself enough credit! :P

    I like in depth because my problem with math and algebra has always been not understanding why, and you helped me understand why in this specific situation. Thanks!
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