Here's a somewhat tough factoring problem if anyone would like to have a go.
Factor $\displaystyle x^{10}+x^{5}+1$
Yes, it factors, though it may take some ingenuity.
Some may not find it that bad, but what the heck, have fun if you wish.
Here's a somewhat tough factoring problem if anyone would like to have a go.
Factor $\displaystyle x^{10}+x^{5}+1$
Yes, it factors, though it may take some ingenuity.
Some may not find it that bad, but what the heck, have fun if you wish.
You can factor $\displaystyle x^2+x+1 = (x - \zeta)(x-\zeta^2)$ where $\displaystyle \zeta = \cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}$ this is what Jhevon did.
That means,
$\displaystyle x^{10}+x^5+1 = (x^5 - \zeta)(x^5 - \zeta^2)$
Now we solve $\displaystyle x^5 - \zeta = 0$. Note that $\displaystyle \zeta^{1/5} = \cos \frac{2\pi}{15}+i\sin \frac{2\pi}{15}$, let us call this number $\displaystyle \eta$. Let $\displaystyle \omega = \cos \frac{2\pi}{5}+i\sin \frac{2\pi}{5}$. That means the solution to $\displaystyle x^5 - \zeta=0$ is $\displaystyle \eta,\eta \omega,\eta \omega^2, \eta \omega^3,\eta \omega^4$.
Now we solve $\displaystyle x^5 - \zeta^2 = 0$. The solutions are: $\displaystyle \eta^2, \eta^2 \omega, \eta^2\omega^2,\eta^2 \omega^3, \eta^2 \omega^4$.
That means the full factorization is:
$\displaystyle \prod_{k=1}^4 (x - \eta \omega^k)(x-\eta^2 \omega^k)$.
yes, that's what i ended up doing in effect. but in getting to that point, i was no where near as elegant. good ol' quadratic formula is what i used
wowThat means,
$\displaystyle x^{10}+x^5+1 = (x^5 - \zeta)(x^5 - \zeta^2)$
Now we solve $\displaystyle x^5 - \zeta = 0$. Note that $\displaystyle \zeta^{1/5} = \cos \frac{2\pi}{15}+i\sin \frac{2\pi}{15}$, let us call this number $\displaystyle \eta$. Let $\displaystyle \omega = \cos \frac{2\pi}{5}+i\sin \frac{2\pi}{5}$. That means the solution to $\displaystyle x^5 - \zeta=0$ is $\displaystyle \eta,\eta \omega,\eta \omega^2, \eta \omega^3,\eta \omega^4$.
Now we solve $\displaystyle x^5 - \zeta^2 = 0$. The solutions are: $\displaystyle \eta^2, \eta^2 \omega, \eta^2\omega^2,\eta^2 \omega^3, \eta^2 \omega^4$.
That means the full factorization is:
$\displaystyle \prod_{k=1}^4 (x - \eta \omega^k)(x-\eta^2 \omega^k)$.
...my factorization looks stupid and incomplete now
I like the clever approaches, but here is what I was getting at. I failed to elaborate. I apologize.
$\displaystyle x^{10}+x^{5}+1$
Here's what I done. I must admit, I pondered it for a while.
Rewrite as $\displaystyle \frac{(x^{5})^{3}-1}{x^{5}-1}$
$\displaystyle =\frac{x^{15}-1}{x^{5}-1}$
$\displaystyle =\frac{(x^{3})^{5}-1}{(x-1)(x^{4}+x^{3}+x^{2}+x+1)}$
$\displaystyle =\frac{(x^{3}-1)(x^{12}+x^{9}+x^{6}+x^{3}+1)}{(x-1)(x^{4}+x^{3}+x^{2}+x+1)}$
$\displaystyle =\frac{(x^{2}+x+1)(x^{12}+x^{9}+x^{6}+x^{3}+1)}{x^ {4}+x^{3}+x^{2}+x+1}$
Now, if we divide:
$\displaystyle \frac{x^{12}+x^{9}+x^{6}+x^{3}+1}{x^{4}+x^{3}+x^{2 }+x+1}$
results in $\displaystyle x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1$
So, we (as Kriz says) happily have:
$\displaystyle (x^{2}+x+1)(x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1)$
There's more than one way to factorize an expression like this. I just thought it was a cool factoring problem.
Yes, I had originally posted something similar to Jhevon's post, but then recalled that there is a theorem of Gauss'(?) that says that any polynomial with integer coefficients can be expressed as a product of linear and quadratic factors with integer coefficients. (This is part of the statement of the Fundamental Theorem of Algebra.) Since there are no real zeros to this function the 10 zeros must be complex and thus represent 5 quadratic factors.
So this means galactus' 8th degree factor has 4 quadratic factors as well... (Too bad I have never learned any general technique to find them!)
-Dan
This my method. Consider the complex equation
$\displaystyle z^{10}+z^5+1=0$
Multiplying by $\displaystyle z^5-1$,
$\displaystyle z^{15}-1=0$
Hence the solutions to $\displaystyle z^{10}+z^5+1=0$ are all the 15th roots of unity minus the 5th roots of unity, i.e.
$\displaystyle z=\cos{\frac{2k\pi}{15}}\pm\mathrm{i}\sin{\frac{2k \pi}{15}},\ k=1,2,4,5,7$
But two of the solutions (with $\displaystyle k=5$) are
$\displaystyle \cos{\frac{2\pi}{3}}\pm\mathrm{i}\sin{\frac{2\pi}{ 3}}=\frac{-1\pm\mathrm{i}\sqrt{3}}{2}$
These are roots of the equation $\displaystyle x^2+x+1=0$. $\displaystyle \fbox{Hence $x^2+x+1$ is a factor of $x^{10}+x^5+1$.}$
Hmmmm... You are right. All those years since I have bothered to apply it had made my brain fuzzy. How about I change that to:
"Every polynomial can be factored (over the real numbers) into a product of linear factors and irreducible quadratic factors."
Apologies and thanks for the catch!
-Dan