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Math Help - How did this factorization come to be?

  1. #1
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    How did this factorization come to be?

    I have not taken algebra in 20 years, and have an algebra class left to complete my associates. Ive been reviewing like crazy, but much of this is still stumping me and I feel behind so I'll be coming in here a bit to see if I can get help, and find a tutor at the college. Im making progress, but its SLOW, and PAINFUL, and time-consuming. Have to do this when my kids are at school in order to have peace to study.


    Anyway, stuck on something here, I'll give you part of the problem below:


    Code:
    3               1                4
    ---      +   ---      =     -----
    X+6          x-2           x(2) + 4x - 12



    The section x(2) + 4x - 12 in the book is show factored to (x+6)(x-2)


    This is done to also find the LCD.


    I dont understand how it got to (x+6)(x-2)??
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  2. #2
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    Re: How did this factorization come to be?

    To multiply (x + 6)(x - 2) through, apply the law of distributivity a(b - c) = ab - ac where a = x + 6, b = x and c = 2. Then apply distributivity two times more.
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  3. #3
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    Re: How did this factorization come to be?

    Thats sort of what I have in my book, but could you visualize each step? I'm somewhat getting it, but I'm still not seeing it fully. Thank you for your response.
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  4. #4
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    Re: How did this factorization come to be?

    One of the most important skills in mathematics is the ability to make a substitution of a concrete expression for a variable into another expression. For example, to substitute a concrete expression x + 6 for a variable a into the law of distributivity a(b - c) = ab - ac, replace a with x + 6. The result is (x + 6)(b - c) = (x + 6)b - (x + 6)c. Then substitute x for b and 2 for c in the equality just obtained.
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  5. #5
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    Re: How did this factorization come to be?

    Quote Originally Posted by itgl72 View Post
    I have not taken algebra in 20 years, and have an algebra class left to complete my associates. Ive been reviewing like crazy, but much of this is still stumping me and I feel behind so I'll be coming in here a bit to see if I can get help, and find a tutor at the college. Im making progress, but its SLOW, and PAINFUL, and time-consuming. Have to do this when my kids are at school in order to have peace to study.


    Anyway, stuck on something here, I'll give you part of the problem below:


    Code:
    3               1                4
    ---      +   ---      =     -----
    X+6          x-2           x(2) + 4x - 12



    The section x(2) + 4x - 12 in the book is show factored to (x+6)(x-2)


    This is done to also find the LCD.


    I dont understand how it got to (x+6)(x-2)??
    Suppose you have a factorised quadratic that looks like \displaystyle \begin{align*} (x + m)(x + n) \end{align*}. When expanded you should get

    \displaystyle \begin{align*} (x + m)(x + n) &= x^2 + n\,x + m\,x + mn \\ &= x^2 + (m + n)x + mn \end{align*}

    Your quadratic looks like this: \displaystyle \begin{align*} x^2 + 4x - 12 \end{align*}. Can you see that it is of the form \displaystyle \begin{align*} x^2 + (m + n)x + mn \end{align*}, with \displaystyle \begin{align*} m + n = 4 \end{align*} and \displaystyle \begin{align*} mn = -12 \end{align*}?

    So to factorise, you are looking for two numbers which multiply to give -12 and add to give 4. If you're good with your rules of numbers and times tables, you will know straight away that the two numbers are 6 and -2.

    So that means \displaystyle \begin{align*} x^2 + 4x - 12 = (x + 6)(x - 2) \end{align*}.
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  6. #6
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    Re: How did this factorization come to be?

    Quote Originally Posted by itgl72 View Post
    The section x(2) + 4x - 12 in the book is show factored to (x+6)(x-2)
    Show x squared this way: x^2.
    Go here, read up, and you'll see how the above "factors":
    Introduction To Factoring -- Algebra.Help
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