My Math Forum Is a complete factorization unique?

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 May 14th, 2015, 02:37 PM #1 Newbie   Joined: Dec 2013 Posts: 28 Thanks: 0 Is a complete factorization unique? Say we are asked to factor a polynomial completely over the integers Is there more than one possible result? for example (3x+3) factors completely as 3(x+1) but if I give as an answer (-3)(-1)(x+1), is it wrong?
 May 14th, 2015, 02:46 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,932 Thanks: 1127 Math Focus: Elementary mathematics and beyond I wouldn't say that it's wrong, but I'd say that it's trivial.
 May 14th, 2015, 04:23 PM #3 Global Moderator   Joined: May 2007 Posts: 6,730 Thanks: 689 When describing unique factorization, factors of 1 don't count. Along the same lines, changing signs of two factors is also not relevant.
 May 14th, 2015, 05:04 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,639 Thanks: 2624 Math Focus: Mainly analysis and algebra Factorisations are unique up to constant multiples and ordering of terms.
May 15th, 2015, 01:45 AM   #5
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The key thing is this phrase...

Quote:
 Originally Posted by mick7 Say we are asked to factor a polynomial completely over the integers
That tells you that the answer must be in its simplest form, which in your case would be $\displaystyle 3(x+1)$. Therefore

Quote:
 (-3)(-1)(x+1), is it wrong?
is wrong because it is not factorised completely. The final result must follow some rules to be considered as factorised completely. V8Archie and Mathman specified what some of those those rules are.

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