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Factoring 4x^4-x^2-18

  1. Nov 4, 2011 #1
    Hello,

    1. The problem statement, all variables and given/known data

    Factor: 4x[itex]^{4}[/itex]-x[itex]^{2}[/itex]-18

    3. The attempt at a solution

    I solved a similar problem x[itex]^{4}[/itex]-6x[itex]^{2}[/itex]+9 by equating x[itex]^{2}[/itex] to t and then reverse-FOIL'ing... this one just wouldn't give in...
    Completing the square also does not help to get the answer (presuming of course that the answer is correct, which I wouldn't dare not to do before consulting in this forum)...

    I have the answear: (x^2+2)(2x-3)(2x+3), so I need your help on the reasoning process guys.
     
    Last edited: Nov 4, 2011
  2. jcsd
  3. Nov 4, 2011 #2

    ehild

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    Replace x2 by t as you did before and complete the square. Then factorize further if it is possible.

    ehild
     
  4. Nov 4, 2011 #3

    Mark44

    Staff: Mentor

    There's also another method for factoring a quadratic in the form ax2 + bx + c. Let u = x2 so that we now have 4u2 - u - 18.

    1. Calculate a*c, which is -72 for this problem.
    2. Find two factors of -72 that add up to -1.
      For this problem, 8 and -9 are factors of -72, and they add to -1.
    3. Rewrite the quadratic with the middle term expanded using the factors found in step 2.
      4u2 - u - 18 = 4u2 + 8u - 9u - 18.
    4. Factor by grouping to get the two binomial factors.
      4u2 + 8u - 9u - 18 = 4u(u + 2) - 9(u + 2) = (4u - 9)(u + 2).

    Don't forget to undo the substitution...
     
  5. Nov 5, 2011 #4
    Thank you
     
  6. Nov 5, 2011 #5

    dextercioby

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    Can't you directly complete the square & then factor ?

    [tex] 4x^4 - x^2 - 18 = \left(2x^2 -\frac{1}{4}\right)^2 - \left(\frac{17}{4}\right)^2 = (2x^2 + 4)(2x^2 - 4.5) [/tex]
     
  7. Nov 5, 2011 #6

    ehild

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    Or [tex](x^2+2)(4x^2-9)=(x^2+2)(2x+3)(2x-3)[/tex]

    ehild
     
  8. Nov 6, 2011 #7

    eumyang

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    This is a great method in factoring quadratic trinomials. I first learned of it in reading Lial's http://www.pearsonhighered.com/educator/product/Introductory-Algebra/9780321557131.page" [Broken] book. It's interesting that when I learned factoring in school we were taught to just guess-and-check. I now teach this method to my freshmen Algebra I classes, even though their books use the guess-and-check method.
     
    Last edited by a moderator: May 5, 2017
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