Question

Find the gcd of$x+a+b$ and $x^3-3abx+a^3+b^3$in $\mathrm{\mathbb{Q}}\left[x\right]$.

Short answer

The gcd of the two polynomial is $\left(x^3-3abx+a^3+b^3,x+a+b\right)=x+a+b$.

Step-by-step solution

Definition of Euclidean algorithm for gcd

Consider the positive integers a and b, where $b\overline{)a}$. Consider that$r_0=a,r_1=b$ and apply the division algorithmrepeatedly to get the set of remainders$r_2,r_3,......,r_n,r_{n+1}$ defined successively by the relations.

role="math" localid="1648620193296" $$\begin{gathered} r_0=r_1{q}_1+r_2,0<r_2<r_1 \\ {r}_1=r_2{q}_2+r_3,0<r_3<r_2 \\ . \\ . \\ . \\ {r}_{n-1}=r_n{q}_n+r_{n+1},r_{n+1}=0 \end{gathered}$$

Then, $r_n$, the last non-zero remainder in this process is known as the gcd of a and b.

Determine the gcd of x+a+b and  x3-3abx+a3+b3in ℚx 

Use the Euclidean algorithm to obtain the gcd of $x+a+b$and $x^3-3abx+a^3+b^3$ as follows:

$x^3-3abx+a^3+b^3=\left(x^2-\left(a+b\right)x+a^2-ab+b^2\right)\left(x+a+b\right)$

As a result, $x+a+b\overline{)x^3-3abx+a^3+b^3}$.

Thus, the gcd of the two polynomial is

$\left(x^3-3abx+a^3+b^3,x+a+b\right)=x+a+b$ .