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Math Help - The art of factoring....

  1. #1
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    The art of factoring....

    I'm a calculus 3 student and the bane of my existence has ALWAYS been factoring. It's just something I constantly struggle with and never learned formally. I'm always BLINDLY taking guesses and "brute forcing" factors with my calculator (which can take well over 25-50 tries easily, and most of the time I never find the answer).

    Is there any way of doing it systematically? Does anyone know a good method? (I'm a calc 3 student, so I don't own an algebra book)

    I can look at any polynomial like

    f(x) = 12x^3-12x^2-24x
    which factors to
    12x(x-2)(x+1)

    But I cannot see why! It looks like magic to me.

    Ok... So I know simple factors like:

    <br />
x^2+3x - 4

    Which comes down to setting up
    (x+a) (x-b)
    And finding something that multiplies to 4 and adds to 3. Those are easy. But honestly, those are about the ONLY ones I can do, and I'm coming across harder ones in my book... Although, I rarely see any that go above a cubic polynomial

    Any advice/help would be appreciated. Thanks.
    Last edited by bobbooey; May 17th 2010 at 06:46 AM.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by bobbooey View Post
    I'm a calculus 3 student and the bane of my existence has ALWAYS been factoring. It's just something I constantly struggle with and never learned formally. I'm always BLINDLY taking guesses and "brute forcing" factors with my calculator (which can take well over 25-50 tries easily, and most of the time I never find the answer).

    Is there any way of doing it systematically? Does anyone know a good method? (I'm a calc 3 student, so I don't own an algebra book)

    I can look at any polynomial like

    f(x) = 3x^4-4x^3-12x^2 + 5
    which factors to
    12x(x-2)(x+1)

    But I cannot see why! It looks like magic to me.

    Ok... So I know simple factors like:

    <br />
x^2+3x - 4

    Which comes down to setting up
    (x+a) (x-b)
    And finding something that multiplies to 4 and adds to 3. Those are easy. But honestly, those are about the ONLY ones I can do, and I'm coming across harder ones in my book... Although, I rarely see any that go above a cubic polynomial

    Any advice/help would be appreciated. Thanks.
    The factorization you have for f(x)=3x^4-4x^3-12x^2+5 is not correct!

    Sometimes a good place to start is to guess the zero of the polynomial. For instance, if x=-1 then f(-1)=3+4-12+5=0. This tells us that x+1 is a factor.

    Next, long divide (or use synthetic division) to reduce the degree of the polynomial. I leave it for you to show that

    \frac{3x^4-4x^3-12x^2+5}{x+1}=3x^3-7x^2-5x+5.

    Also observe that if x=-1, then 3(-1)^3-7(-1)^2-5(-1)+5=-3-7+5+5=0; therefore, x+1 is a factor of 3x^3-7x^2-5x+5. Apply long division again to get

    \frac{3x^3-7x^2-5x+5}{x+1}=3x^2-10x+5.

    Therefore, 3x^4-4x^3-12x^2+5=(3x^2-10x+5)(x+1)^2.

    Observe that 3x^2-10x+5 cannot be factored nicely (it will involve square roots), so its best that we leave it like I did above.

    Factoring in general is a game (to me anyways) and requires some intuition; there are a couple theorems out there that help make it a little easier (like the method I just used). Maybe you should take a look at those.

    Does this make some sense?

    EDIT: Take a look at this: http://tutorial.math.lamar.edu/Class...Factoring.aspx and see if this helps demystify the factoring process a little.
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