求問二題

1.f(x)=x^34-3x^20+11x-7除以x^2-x+1的餘式?
2.設x^99+x^30+2除以(x^2+1)(x^2-1)的餘式為ax^3+bx^2+cx+d除以x^2+1的餘式為r1(x)，除以x^2-1的餘式為r2(x) 求r1(x)+r2(x)=?
謝謝老師們

(1)
利用\(x^2-x+1)(x+1)=x^3+1\)
再利用\(x^3\)除以\(x^3+1\)的餘式為\(-1\)
則\(x^{34}=(x^3)^{11} x\)除以\(x^3+1\)的餘式為\( (-1)^{11}=-x x\)
\(-3x^{20}=(x^3)^{6} x^2\)除以\(x^3+1\)的餘式為\( 3(-1)^{6} x^2=-3x^2\)
所以\(x^{34}-3x^{20}+11x-7\)除以\(x^3+1\)的餘式為\(-x-3x^2+11x-7=-3x^2+10x-7\)
而\(x^{34}-3x^{20}+11x-7\)除以\(x^2-x+1\)的餘式為\(-3x^2+10x-7\)除以\(x^2-x+1\)的餘式
所以答案為\(7x-4\)