This is one possible way to code the EXTENDED EUCLIDEAN ALGORITHM (for your homework you need to have a few extra auxiliary variables.f:=824*x^5-65*x^4-814*x^3-741*x^2-979*x-764;

NiM+SSJmRzYiLC4qJEkieEdGJSIiJiIkQykqJEYoIiIlISNsKiRGKCIiJCEkOSkqJEYoIiIjISRUKEYoISR6KiEkaygiIiI=

g:=216*x^4+663*x^3+880*x^2+916*x+617;

NiM+SSJnRzYiLCwqJEkieEdGJSIiJSIkOyMqJEYoIiIkIiRqJyokRigiIiMiJCEpKUYoIiQ7KiIkPCciIiI=

rho[0]:=lcoeff(f,x);

NiM+JkkkcmhvRzYiNiMiIiEiJEMp

r[0]:=f/rho[0];

NiM+Jkkickc2IjYjIiIhLC4qJEkieEdGJiIiJiIiIiokRisiIiUjISNsIiRDKSokRisiIiQjISQyJSIkNyUqJEYrIiIjIyEkVChGMkYrIyEkeipGMiMhJCI+IiQxI0Yt

rho[1]:=lcoeff(g,x); r[1]:=g/rho[1];

NiM+JkkkcmhvRzYiNiMiIiIiJDsjNiM+Jkkickc2IjYjIiIiLCwqJEkieEdGJiIiJUYoKiRGKyIiJCMiJEAjIiNzKiRGKyIiIyMiJDUiIiNGRisjIiRIIyIjYSMiJDwnIiQ7I0Yo

printlevel:=2;

NiM+SStwcmludGxldmVsRzYiIiIj

for i from 1 to 5 do q[i]:=quo(r[i-1],r[i],x,'a[i+1]'); a[i+1]:=sort(a[i+1]); if (a[i+1] <> 0) then rho[i+1]:=lcoeff(a[i+1],x); r[i+1]:=a[i+1]/rho[i+1];fi;od;

NiM+JkkicUc2IjYjIiIiLCZJInhHRiZGKCMhJVBlIiVhPUYoNiM+JkkiYUc2IjYjIiIjLCoqJEkieEdGJiIiJCMiJ3BVaCInKVtMIiokRitGKCMiKCYzUjoiJ0stP0YrIyInWDwkKiInOyw1IyIoRCxCJCInay9TIiIiNiM+JkkkcmhvRzYiNiMiIiMjIidwVWgiJylbTCI=NiM+Jkkickc2IjYjIiIjLCoqJEkieEdGJiIiJCIiIiokRitGKCMiKHEieUkiKDJHJT1GKyMiKCEpcHMkRjEjIihELEIkRjFGLQ==NiM+JkkicUc2IjYjIiIjLCZJInhHRiYiIiIjIilwdCg9JyIpb3RBV0YrNiM+JkkiYUc2IjYjIiIkLCgqJEkieEdGJiIiIyMhLjAjXCo9P0giIi5LQiZvInpfJUYrIyEtRkROSm5RIi4kMzgjej44IiMiLW5fVGxcIioiLm1oVWVSRSMiIiI=NiM+JkkkcmhvRzYiNiMiIiQjIS4wI1wqPT9IIiIuS0ImbyJ6XyU=

NiM+Jkkickc2IjYjIiIkLCgqJEkieEdGJiIiIyIiIkYrIyIuMyxURHBhIiIuMCNcKj0/SCIjIS5NMCQzJCpIPUYwRi0=NiM+JkkicUc2IjYjIiIkLCZJInhHRiYiIiIjIjQlcGglXFkqKm9qNyIiNE4lPXdzajolNFEjRis=NiM+JkkiYUc2IjYjIiIlLCZJInhHRiYjIjpNczM7K1ZpZClRKVt6JSI6RD84cUJ6L14nSEpwOyMiOi4qNCYqKj1zUiJSJT1YLyVGLSIiIg==NiM+JkkkcmhvRzYiNiMiIiUjIjpNczM7K1ZpZClRKVt6JSI6RD84cUJ6L14nSEpwOw==NiM+Jkkickc2IjYjIiIlLCZJInhHRiYiIiIjIjNWPS4pcFtsZUooIjNhRF1tekQ6dCcpRis=NiM+JkkicUc2IjYjIiIlLCZJInhHRiYiIiIjIj88NSkpKlFHWzY5OUFGJFtrUiJAcSZwKEchM1w4V3RBW3hlPzZGKw==

NiM+JkkiYUc2IjYjIiImIyFGJlwoSCZldnhPMyQzJylmIjNHLiEqKkciIkU7SF8pKUclKmY5Jz4oM10sbnZOQXY=NiM+JkkkcmhvRzYiNiMiIiYjIUYmXChIJmV2eE8zJDMnKWYiM0cuISoqRyIiRTtIXykpRyUqZjknPigzXSxudk5Bdg==NiM+Jkkickc2IjYjIiImIiIiNiM+JkkicUc2IjYjIiImLCZJInhHRiYiIiIjIjNWPS4pcFtsZUooIjNhRF1tekQ6dCcpRis=NiM+JkkiYUc2IjYjIiInIiIh

MAPLE contains a nice fast subroutine to compute the extended GCD algorithmgcdex(f,g,x,'s','t');

NiMiIiI=

s;

NiMsKiMhNl9UJT0/dD1ENnl4IjlSeEdfTHc9V0FnJCkqKiIiIkkieEc2IiMhNig9c29UKilmQ0pGdEYmKiRGKCIiIyMhNjlfSjtJXkB0KzwlRiYqJEYoIiIkIyE2aztiazJzJjRTdD1GJg==

t;

NiMsLCMiNiRHJFxuOl47TSdcbCI5UnhHX0x3PVdBZyQpKioiIiJJInhHNiIjITc3OmReVmRxYiNRNiRGJiokRigiIiMjIjcuUHdEUSMzLkdiMiJGJiokRigiIiQjITZVWSE0KyYqeSFHP2YnRiYqJEYoIiIlIyI2J1wvIilSa0R4bllyRiY=

s*f+t*g;

NiMsJiomLCojITZfVCU9P3Q9RDZ5eCI5UnhHX0x3PVdBZyQpKioiIiJJInhHNiIjITYoPXNvVCopZkNKRnRGKCokRioiIiMjITY5X0o7SV5AdCs8JUYoKiRGKiIiJCMhNms7YmsycyY0U3Q9RihGKSwuKiRGKiIiJiIkQykqJEYqIiIlISNsRjIhJDkpRi4hJFQoRiohJHoqISRrKEYpRilGKSomLCwjIjYkRyRcbjpeO00nXGxGKEYpRiojITc3OmReVmRxYiNRNiRGKEYuIyI3LlB3RFEjMy5HYjIiRihGMiMhNlVZITQrJip5IUc/ZidGKEY6IyI2J1wvIilSa0R4bllyRihGKSwsRjoiJDsjRjIiJGonRi4iJCEpKUYqIiQ7KiIkPCdGKUYpRik=

simplify(%);

NiMiIiI=