September 2nd, 2014, 01:54 AM  #1 
Newbie Joined: Jun 2014 From: UK Posts: 20 Thanks: 0  Factoring Polynomials
Hello, is there an "easy" way to find nonrational factors of polynomials? What I mean is, if I have a polynomial such as: $\displaystyle x^2+2x15$ ...it's not too hard to figure out that it factors to: $\displaystyle (x3)(x+5)$ ...but, if I have a polynomial such as: $\displaystyle x^2+9x+9$ ...then the factors are much harder to find. I've written a piece of software to find them by brute force (a loop inside a loop that stops when the two loop variables equal the two coefficients in the polynomial), but I'm currently running it on the polynomial above and it's not showing any signs of success yet <edit> it finished while I was typing this and I accidentally pressed enter and cleared the result  now I'm having to run it again  aargh! <edit> I know the solution for the above polynomial is: $\displaystyle (x+7.854)(x+1.146)$ ...but how do I get there? Any suggestions will be very much appreciated, not least by my laptop cpu... Thanks 
September 2nd, 2014, 08:23 AM  #2 
Newbie Joined: Jun 2014 From: UK Posts: 20 Thanks: 0 
...I thought I should probably just say that I realise I can find the roots of a quadratic polynomial via $\displaystyle \frac{b \pm \sqrt{b^24ac}}{2a}$ but I was looking for a factoring way of working with nonrational numbers, if you see what I mean...

September 2nd, 2014, 08:35 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,972 Thanks: 2293 Math Focus: Mainly analysis and algebra 
You could try completing the square, but that is really the same as $$x = \tfrac{1}{2a}(b \pm \sqrt{b^2  4ac})$$ For what it's worth: \begin{align*} x^2 + 9x + 9 &= (x + \tfrac92)^2 + (9\tfrac{81}{4}) = 0 \\ (x + \tfrac92)^2 &= \tfrac{45}{4} \\ x + \tfrac92 &= \pm \sqrt{\tfrac{45}{4}} \\ x &= \tfrac92 \pm \tfrac32 \sqrt{5} \\ \end{align*} Last edited by v8archie; September 2nd, 2014 at 08:37 AM. Reason: Added the radical symbol 
September 2nd, 2014, 09:09 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
What degrees of polynomials are you working with? The answer will be very different if you say "2" than if you say "1000".


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