# Every polynomial in $\mathbb{Q}[x]$ is a product of a constant polynomial and a primitive polynomial.

Definition: A polynomial $$f\in \mathbb{Z}[x]$$ with $$f(x)=a_0+a_1x+\dots+a_nx^n$$ is a primitive polynomial the only common factors of $$a_0,a_1,\dots,a_n$$ are the units $$\pm 1\in \mathbb{Z}$$ and if $$a_n>0$$.

I stumbled across this sentence in my course notes, but I can't find a reasonable explanation for it:

Every polynomial $$f\in\mathbb{Q}[x]$$ can be uniquely written as a product $$f(x)=c\cdot f_0(x)$$ with $$c\in\mathbb{Q}$$ and $$f$$ a primitive polynomial in $$\mathbb{Z}[x]$$.

Thanks.

• Not quite: if $f$ is primitive, so is $-f$. So, for uniqueness you'd have to insist that $c$ is a positive rational. – Angina Seng Dec 24 '19 at 13:52

We can multiply all the coefficients of $$f$$ by an integer so that they are all integers and so that the coefficient of the highest power of $$x$$ is positive.Then any common factor of all the coefficients can be factored out. The effect of this is that a rational times $$f$$ is a primitive polynomial.
If the ratio of two primitive polynomials were a rational, $$a/b$$ say, then let the polynomial $$g$$ be $$a$$ times one and also $$b$$ times the other. Then the g.c.d. of all the terms of $$g$$ would be $$a$$ and also would be $$b$$. Therefore $$a=b$$ and the representation is unique.
• Or $a=-b{}{}{}$? – Angina Seng Dec 24 '19 at 14:02
• No - see the definition of primitive with $a_n$ positive. – S. Dolan Dec 24 '19 at 14:03