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Thread: 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.

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Example: $\displaystyle 6x^2 - 35x + 36$

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

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


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

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

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

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

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

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

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


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

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

    Answer: .$\displaystyle (3x - 4)(2x - 9)$

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    Example: .$\displaystyle 18x^2 - 11x - 24$

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


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


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

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

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

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


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

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

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

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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, $\displaystyle x^2 - 9$ or $\displaystyle 2x^2 + 6x$

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  2. #2
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    It reminds me of the "box method"

    If you have a trinomial, $\displaystyle ax^2+bx+c$ 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 $\displaystyle 2x^2+5x+3$ 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: $\displaystyle 2x^2+5x+3=(2x+3)(x+1)$
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