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I was attempting to factor the following expression when I realized I didn't even know what type of function the expression is. Does anyone know? $x^4-9x^x+12x-4$.

Sudarsan
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user97722
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    Ugly. I'm going with ugly. That, or typo. – Cameron Buie Sep 29 '13 at 22:51
  • May I ask from which course this question is? – imranfat Sep 29 '13 at 23:11
  • It's from Precalc, but my teacher likes to give these kinds of convoluted questions each week as an extra challenge for students who are interested. I mostly do them out of interest. – user97722 Sep 29 '13 at 23:29
  • I'm been puzzling over it for the whole weekend here is what I've come up with so far. As far as I know the x^x is basically Knuth's up arrow notation as in x^2=x↑↑2, which is the same notation used to express Graham's number. If the equation was just 9x^x-4 then you could factor it as (3\sqrt(x)^x-2)(3\qrt(x)+2), but sadly that is not the answer. I don't know if that helps, but that's all I got so far. – user97722 Sep 29 '13 at 23:33
  • Not all functions are named, or belong to a named class. – Michael Albanese Sep 30 '13 at 00:07
  • Are there anyways to factor this function then? Since it's neither polynomial nor exponential, which way do you factor it? – user97722 Sep 30 '13 at 00:14
  • If your instructor provides an answer, please post it here (you can answer your own question). I'd love to see it. It's exasperating, since I've come up with solutions that almost work. In particular, one can use the fact that $x^x-x^2$ and $x^{x/2}-x$ both have roots $x=1, 2$, so could be factors. – Rick Decker Oct 01 '13 at 00:54

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If it's a typo and should actually be $x^4-9x^2+12x-4$, that has a particularly nice factorization: $$\begin{align} x^4-9x^2+12x-4&=(x-1)(x^3+x^2-8x+4)&\quad\text{since $x=1$ is a root}\\ &=(x-1)(x-2)(x^2+3x-2)&\quad\text{since $x=2$ is a root}\\ &=(x^2-3x+2)(x^2+3x-2) \end{align}$$ or, if you prefer, $$ x^4-9x^2+12x-4=x^4-(3x-2)^2 $$ In a precalc course, it's unlikely that you'd see anything like $x^x$ as part of an equation you'd have to solve, so the default would be to assume a typo (or thinko, which I do more often than I'd like).

Update. It's worth mentioning that $x=1, x=2$ are also roots of $$ x^4-9x^x+12x-4 $$

Rick Decker
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  • Thanks for the response, but I do not believe it is a typo. The question is suppose to be challenging. However, I am also equally unsure if there is a factorization. At this point I'm leaning towards there is no possible factorization, but it would be nice to see some definitive proof just out of curiosity. – user97722 Sep 30 '13 at 01:15
  • I have looked into factorization of a polynomial over a finite field, but I haven't come up with a conclusive answer yet. I'll continue trying for maybe a few more hours then I'll probably call it a day and try again some other time. – user97722 Sep 30 '13 at 01:17
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    @user97722 if its not a typo, then the given function is definitely not a polynomial. – Gamma Function Sep 30 '13 at 01:58