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Math Help - How does this simplify to this?

  1. #1
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    How does this simplify to this?

    So I was just doing a calculus question and on the solutions ..

    it jumps from

    \frac{3(x+2)^2(x^2+1)-2x(x+2)^3}{(x^2+1)^2}

    then it skips to

    \frac{(3(x^2+1)-2x(x+2))(x+2)^2}{(x^2+1)^2}

    and I don't know how that works.. there have been a number of questions that have done that and now I think its time i asked about it. Of course, thats not the simplest form but I understand how it simplifies from there.. I just don't get how it skips from the first one to the second.

    If someone could care to explain I'd really love it thanks
    Last edited by Klutz; December 18th 2010 at 10:22 PM.
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  2. #2
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    Are you sure it's not \displaystyle \frac{3(x+2)^2(x^2+1) - 2x(x+2)^3}{(x^2+1)^2}?

    If it is, they've realised there's a common factor of \displaystyle (x+2)^2 in the numerator.
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  3. #3
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    sorry yeah i'll fix it my bad!
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  4. #4
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    I still don't get it though.. could you explain in more depth?
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  5. #5
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    \displaystyle 3(x+2)^2(x^2+1) - 2x(x+2)^3 = 3(x+2)^2(x^2+1) - 2x(x+2)(x+2)^2.

    Do you see that there is a common factor of \displaystyle (x+2)^2?
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  6. #6
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    Yeah I see it but why is it that there are 5 of them in the first equation and only 3 in the 2nd one?
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  7. #7
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    ab+ ac= a(b+ c) has, in your sense, "two" a's on the left and only one on the right!

    6+ 10= 2(3)+ 2(5)= 2(3+ 5) has two "2"s on the left and only one on the right. They are not supposed to be the same!
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  8. #8
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    I guess that makes sense.. still don't completely understand though D:
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  9. #9
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    Hello, Klutz!

    \dfrac{3(x+2)^2(x^2+1)-2x(x+2)^3}{(x^2+1)^2} \;=\;\dfrac{(x+2)^2\bigg[3(x^2+1)-2x(x+2)\bigg]}{(x^2+1)^2}

    Examine the numerator: . 3(x+2)^2(x^2+1) - 2x(x+2)^3


    The two terms have a common factor: . 3\underbrace{(x+2)^2}(x^2+1) \;-\; 2x\underbrace{(x+2)^3}
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _{\text{They have a common factor of }(x+2)^2}

    Factor it out: . (x+2)^2\,\bigg[3(x^2+1) - 2x(x+2)\bigg]

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  10. #10
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    ahh! I got it thanks so much. Gosh I'm so slow.
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