Factoring 4x^4-x^2-18

1. Nov 4, 2011

mindauggas

Hello,

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

Factor: 4x$^{4}$-x$^{2}$-18

3. The attempt at a solution

I solved a similar problem x$^{4}$-6x$^{2}$+9 by equating x$^{2}$ 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. Nov 4, 2011

ehild

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

ehild

3. Nov 4, 2011

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...

4. Nov 5, 2011

Thank you

5. Nov 5, 2011

dextercioby

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

$$4x^4 - x^2 - 18 = \left(2x^2 -\frac{1}{4}\right)^2 - \left(\frac{17}{4}\right)^2 = (2x^2 + 4)(2x^2 - 4.5)$$

6. Nov 5, 2011

ehild

Or $$(x^2+2)(4x^2-9)=(x^2+2)(2x+3)(2x-3)$$

ehild

7. Nov 6, 2011

eumyang

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