Find the g.c.d. of each of the following pairs of polynomials
\(2x^5 - x^3 - 3x^2 - 6x + 4\), \(\;x^4 + x^3 - x^2 - 2x - 2\);
\(3x^4 + 8x^2 - 3\), \(\;x^3 + 2x^2 + 3x + 6\);
\(x^4 - 2x^3 - 2x^2 - 2x - 3\), \(\;x^3 + 6x^2 + 7x + 1\).
Solution
For each pair, apply the Euclidean algorithm over \(\QQ[x]\) and normalize the last nonzero remainder to be monic.
For \[ f=2x^5-x^3-3x^2-6x+4, \qquad g=x^4+x^3-x^2-2x-2, \] the Euclidean algorithm terminates with a nonzero constant remainder. Hence \[ \gcd(f,g)=1. \]
For \[ f=3x^4+8x^2-3, \qquad g=x^3+2x^2+3x+6, \] one finds \[ f=(3x-6)g+11(x^2+3), \] and \[ g=(x+2)(x^2+3). \] Therefore \[ \gcd(f,g)=x^2+3. \]
For \[ f=x^4-2x^3-2x^2-2x-3, \qquad g=x^3+6x^2+7x+1, \] the Euclidean algorithm again terminates with a nonzero constant remainder. Hence \[ \gcd(f,g)=1. \]
Thus the three monic gcds are \[ \boxed{1,\quad x^2+3,\quad 1}. \]