Problem Concerning The Euclidean Algorithm For Polynomials

Let $P(x) = x^3-x^2-x-2$ and $Q(x) = 2x^2-3x-2$

What do I do in the first step when The coefficient in front of $2x^2$ is greater than the coefficient in front of $x^3$ ($2>1$)

In the first step do I simply write $x^3 = \dfrac{1}{2}2x^2\times x$ And do I continue with non-whole coefficients?

Or is therea way to always have whole coefficients?

The ring of polynomials with integer coefficients, $\mathbb{Z}[X]$, is not an Euclidean domain, which means that the Euclidean division algorithm is not available in it. But since the ring of polynomials with rational coefficients is Euclidean, we can actually divide if we allow for rational coefficients. For example, if you divide $X^2$ by $2X$ you necessarily get $\frac12 X$.
A related, but different question, is if given two polynomials with integer coefficients there always exists a greatest common divisor for them, and the answer is yes. For the previous example we have $\gcd(X^2,2X)=X$.
• So If I am in the $\mathbb{Q}[x]$ I can have non-whole coefficients?! – aribaldi Jun 20 '16 at 15:21
• @aribaldi Yes, that's it: $\mathbb{Q}[X]$ means polynomials with coefficients in $\mathbb{Q}$, which by definition is the set of rational numbers, i.e., fractions of the form $p/q$ with $p,q$ integers and $q\neq 0$, which are not whole numbers. – Jose Brox Jun 20 '16 at 15:33