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Math Help - Factoring trinomials

  1. #1
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    Factoring trinomials

    This method of factoring trinomials was shown to me
    . . by one of my students many years ago.
    You may find it as fascinating as I did.

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    Example: 6x^2 - 35x + 36

    Write the two pairs of parentheses and "split the x's": . (\;x\qquad)(\:x\qquad)

    Use the first coefficient twice: . (\overbrace{6x}^\downarrow\qquad)(\overbrace{6x}^\  downarrow\qquad)


    Multiply the first coefficient by the last coefficient: .  6 \times 36 \,=\,216

    We will factor 216 into two parts.
    Note the sign of the last term.
    . . If "+", think sum.
    . . If "-", think difference.

    Since the last sign is "+", factor 216 into two parts
    . . whose sum is the middle coefficient (35).

    To factor 216 into all possible pairs, divide by 1, 2, 3, ...
    . . keeping those that "come out even".

    216\:=\:\begin{Bmatrix}1\cdot216 \\2\cdot108\\3\cdot72 \\ 4\cdot54\\6\cdot36\\8\cdot27\\9\cdot24\\12\cdot18\  end{Bmatrix}

    The pair with a sum of 35 is: 8 and 27.

    Since the middle coefficient is -35, we will use: -8 and -27.


    Insert them into the parentheses: . (6x \overbrace{-\, 8}^\downarrow)(6x \overbrace{-\, 27}^\downarrow)

    Factor out all common factors and discard them: . \not{2}(3x - 4)\not{3}(2x - 9)

    Answer: . (3x - 4)(2x - 9)

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    Example: . 18x^2 - 11x - 24

    We have: . (\underbrace{18x}\qquad)(\underbrace{18x}\qquad)


    Then: . 18 \times 24 \,= \,432


    The last term is negative, think difference.
    Factor 432 into two factors whose difference is 11.

    432 \:=\:\begin{Bmatrix}1\cdot432 \\ 2\cdot216\\3\cdot144\\ \vdots \\ 16\cdot27\\18\cdot24\end{Bmatrix}

    The pair with a difference of 11 is 16 and 27.

    Since we want -11, we will use: +16 and -27.


    Insert them into the parentheses: . (18x \underbrace{+\, 16})(18x \underbrace{-\, 27})

    Factor out common factors and discard them: . \not{2}(9x+ 8)\not{9}(2x-3)

    Answer: . (9x + 8)(2x - 3)

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    At this point, someone will say, "Why not use the Quadratic Formula?"
    . . Why not, indeed?
    In fact, why bother teaching Factoring at all?
    . . And I don't have an answer to that question.

    But it would be a shame to run through the Quadratic Formula
    . . to factor, say, x^2 - 9 or 2x^2 + 6x

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  2. #2
    MHF Contributor Quick's Avatar
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    It reminds me of the "box method"

    If you have a trinomial, ax^2+bx+c then write out a box like this:
    Code:
    :-----:-----:
    :  a  :     :
    :-----:-----:
    :     :  c  :
    :-----:-----:
    Now you need to fill in the two empty squares. The trick is that they add to "b" and multiply to "ac", I'll use 2x^2+5x+3 as an example:

    Code:
    :-----:-----:
    :  2  :  2  :
    :-----:-----:
    :  3  :  3  :
    :-----:-----:
    Then find the greatest common factor of each row and collum:

    Code:
         1      1
      :-----:-----:
    2 :  2  :  2  :
      :-----:-----:
    3 :  3  :  3  :
      :-----:-----:
    So: 2x^2+5x+3=(2x+3)(x+1)
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