Finding the $\gcd$ of polynomials in $\Bbb R[x]$

Question

Let $f(x)=6x^3-10x^2-6x+10$ and $g(x)=3x^2-14x+15$ in $\Bbb R[x]$. I want to find the $\gcd$ of these two polynomials.

I am not really sure how to do this in general, but my approach was as follows:

Using basic algebra, the zeros of $g$ are $3$ and $5/3$. Thus $g(x)=(x-3)(x-5/3)$. Then I computed $f(3)\neq 0$ and $f(5/3)=0$. So $f$ is of the form $f(x)=(x-5/3)h(x)$ with $h$ being of degree $2$.

Then the $\gcd$ is $(x-5/3)$. Is that true?

Answer 1

Your strategy is right but not general. And $g(x)=3(x-3)(x-5/3)=(x-3)(3x-5)$.

In general case, you can do long division and extended Euclidean algorithm:

$ 6x^3-10x^2-6x+10 \equiv 48x-80 \pmod{3x^2-14x+15}$

$gcd(f(x),g(x)) = gcd(g(x),48x-80) = gcd((x-3)(3x-5),16(3x-5)) = 3x-5$ as desired.

EDIT: As Bernard suggests, the great common divisional polynomial should be monic due to convention so $3x-5$ is not the final answer and $gcd(f(x),g(x))$ = $x-5/3$ by dividing the coefficient of first non-zero term.