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May 14th, 2015, 03:37 PM   #1
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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?
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May 14th, 2015, 03:46 PM   #2
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I wouldn't say that it's wrong, but I'd say that it's trivial.
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May 14th, 2015, 05:23 PM   #3
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When describing unique factorization, factors of 1 don't count. Along the same lines, changing signs of two factors is also not relevant.
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May 14th, 2015, 06:04 PM   #4
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Factorisations are unique up to constant multiples and ordering of terms.
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May 15th, 2015, 02:45 AM   #5
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The key thing is this phrase...

Quote:
Originally Posted by mick7 View Post
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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