1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Factoring 4x^4-x^2-18

  1. Nov 4, 2011 #1

    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


    User Avatar
    Homework Helper

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

  4. Nov 4, 2011 #3


    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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Homework Helper

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

  8. Nov 6, 2011 #7


    User Avatar
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook