batch("H:/XXXXXXXXXXXXX/Prax15.txt");

/*   265a   */

f: 1*((x+1)*(x-%i)*(x+%i))/((1+1)*(1-%i)*(1+%i)) + 
   2*((x-1)*(x+1)*(x+%i))/((%i-1)*(%i+1)*(%i+%i)) + 
   3*((x-1)*(x-%i)*(x+%i))/((-1-1)*(-1-%i)*(-1+%i)) +
   4*((x-1)*(x+1)*(x-%i))/((-%i-1)*(-%i+1)*(-%i-%i));

ratsimp(expand(f));

f: x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1;
df: diff(f,x);

f1: ratsimp(9*second(divide(f,df)));
f2: ratsimp(1/144*second(divide(df,f1)));
f3: ratsimp(second(divide(f1,f2)));

g: ratsimp(first(divide(f,f2)));

factor(g);

factor(g)*factor(f2);

factor(f);





/*   265c   */

f: x^6-15*x^5+72*x^4-95*x^3-72*x^2-15*x-1;
df: diff(f,x);

f1: ratsimp(-4/29*second(divide(f,df)));
f2: ratsimp(second(divide(df,f1)));

g: ratsimp(first(divide(f,f1)));

solve(g = 0,x);

factor(g)*factor(f1);

factor(f);





/*   266d   */

f: (x+2)^2;
g: (x-1)^2*x;

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

ratsimp(expand(a*(x-1)^2 + (c*x+d)*x)) = f1;

A: matrix(
[1,1,0,1],
[-2,0,1,4],
[1,0,0,4]
);

echelon(A);

l: linsolve([
1*x+1*y+0*z=1,
-2*x+0*y+1*z=4,
1*x+0*y+0*z=4
],[x,y,z]);

p2: ratsimp(second(divide(ev(x,first(l)),x)));

p3: ratsimp(second(divide(ev(y,second(l))*x+ev(z,third(l)),x-1)));
p4: ratsimp(first(divide(ev(y,second(l))*x+ev(z,third(l)),x-1)));

v: p4/(x-1)+p3/(x-1)^2+p2/x+p1;

partfrac(f/g,x);






/*   266f   */

f: x^4+2;
g: x^2*(x+1)^2;

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

ratsimp(expand((a*x+b)*(x+1)^2 + (c*x+d)*(x)^2)) = f1;


A: matrix(
[1,0,1,0,-2],
[2,1,0,1,-1],
[1,2,0,0,0],
[0,1,0,0,2]
);

echelon(A);

l: linsolve([
1*x+0*y+1*z+0*u=-2,
2*x+1*y+0*z+1*u=-1,
1*x+2*y+0*z+0*u=0,
0*x+1*y+0*z+0*u=2
],[x,y,z,u]);

p2: ratsimp(second(divide(ev(x,first(l))*x+ev(y,second(l)),x)));
p3: ratsimp(first(divide(ev(x,first(l))*x+ev(y,second(l)),x)));

p4: ratsimp(second(divide(ev(z,third(l))*x+ev(u,fourth(l)),x+1)));
p5: ratsimp(first(divide(ev(z,third(l))*x+ev(u,fourth(l)),x+1)));

v: p5/(x+1)+p4/(x+1)^2+p3/x+p2/x^2+p1;

partfrac(f/g,x);





/*   266j   */

f: x^3;
g:x^2+4*x+8;
l0: solve(g=0);

g: coeff(g,x,hipow(g,x))*(x-ev(x,first(l0)))*(x-ev(x,second(l0)));

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

ratsimp(expand((a)*(x-ev(x,first(l0))) + (b)*(x-ev(x,second(l0)))) = f1);

A: matrix(
[1,1,8],
[2+2*%i,2-2*%i,32]
);

echelon(A);

l: linsolve([
1*x+1*y=8,
(2+2*%i)*x+(2-2*%i)*y=32
],[x,y]);

p2: ratsimp(second(divide(ev(x,first(l)),(x-ev(x,second(l0))))));
p3: ratsimp(second(divide(ev(y,second(l)),(x-ev(x,first(l0))))));

v: p3/(x-ev(x,first(l0)))+p2/(x-ev(x,second(l0)))+p1;

partfrac(f/g,x);




/*   267b   */

f: 1;
g:x^4+4;
l0: solve(g=0);

g1: (x-realpart(ev(x,first(l0))) - imagpart(ev(x,first(l0)))*%i);
g2: (x-realpart(ev(x,second(l0))) - imagpart(ev(x,second(l0)))*%i);
g3: (x-realpart(ev(x,third(l0))) - imagpart(ev(x,third(l0)))*%i);
g4: (x-realpart(ev(x,fourth(l0))) - imagpart(ev(x,fourth(l0)))*%i);

g: coeff(g,x,hipow(g,x))*g1*g2*g3*g4;

ratsimp(g);

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

l1: ratsimp(expand((a)*g1*g2*g3 + (b)*g1*g2*g4 + (c)*g1*g3*g4 + (d)*g2*g3*g4 ));

A: matrix(
[coeff(coeff(l1,x,3),a),coeff(coeff(l1,x,3),b),coeff(coeff(l1,x,3),c),coeff(coeff(l1,x,3),d),coeff(f1,x,3)],
[coeff(coeff(l1,x,2),a),coeff(coeff(l1,x,2),b),coeff(coeff(l1,x,2),c),coeff(coeff(l1,x,2),d),coeff(f1,x,2)],
[coeff(coeff(l1,x,1),a),coeff(coeff(l1,x,1),b),coeff(coeff(l1,x,1),c),coeff(coeff(l1,x,1),d),coeff(f1,x,1)],
[coeff(coeff(l1,x,0),a),coeff(coeff(l1,x,0),b),coeff(coeff(l1,x,0),c),coeff(coeff(l1,x,0),d),coeff(f1,x,0)]
);

echelon(A);

l: linsolve([
A[1,1]*x+A[1,2]*y+A[1,3]*z+A[1,4]*u=A[1,5],
A[2,1]*x+A[2,2]*y+A[2,3]*z+A[2,4]*u=A[2,5],
A[3,1]*x+A[3,2]*y+A[3,3]*z+A[3,4]*u=A[3,5],
A[4,1]*x+A[4,2]*y+A[4,3]*z+A[4,4]*u=A[4,5]
],[x,y,z,u]);


p2: ratsimp(second(divide(ev(x,first(l)),g4)));
p3: ratsimp(second(divide(ev(y,second(l)),g3)));
p4: ratsimp(second(divide(ev(z,third(l)),g2)));
p5: ratsimp(second(divide(ev(u,fourth(l)),g1)));

v: p5/g1+p4/g2+p3/g3+p2/g4+p1;


partfrac(f/g,x);
realpart(-1/(8*%i-8))+imagpart(-1/(8*%i-8))*%i;
realpart(1/(-8*%i-8))+imagpart(1/(-8*%i-8))*%i;
realpart(-1/(-8*%i-8))+imagpart(-1/(-8*%i-8))*%i;
realpart(1/(8*%i-8))+imagpart(1/(8*%i-8))*%i;


/*   267c   */

f: 3+x;
g:(x-1)*(x^2+1);
l0: solve(g=0);

g1: (x-realpart(ev(x,first(l0))) - imagpart(ev(x,first(l0)))*%i);
g2: (x-realpart(ev(x,second(l0))) - imagpart(ev(x,second(l0)))*%i);
g3: (x-realpart(ev(x,third(l0))) - imagpart(ev(x,third(l0)))*%i);


g: coeff(expand(g),x,hipow(expand(g),x))*g1*g2*g3;

ratsimp(g-(x-1)*(x^2+1));

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

l1: ratsimp(expand((a)*g1*g2 + (b)*g1*g3 + (c)*g2*g3 ));

A: matrix(
[coeff(coeff(l1,x,2),a),coeff(coeff(l1,x,2),b),coeff(coeff(l1,x,2),c),coeff(f1,x,2)],
[coeff(coeff(l1,x,1),a),coeff(coeff(l1,x,1),b),coeff(coeff(l1,x,1),c),coeff(f1,x,1)],
[coeff(coeff(l1,x,0),a),coeff(coeff(l1,x,0),b),coeff(coeff(l1,x,0),c),coeff(f1,x,0)]
);

echelon(A);

l: linsolve([
A[1,1]*x+A[1,2]*y+A[1,3]*z=A[1,4],
A[2,1]*x+A[2,2]*y+A[2,3]*z=A[2,4],
A[3,1]*x+A[3,2]*y+A[3,3]*z=A[3,4]
],[x,y,z]);


p2: ratsimp(second(divide(ev(x,first(l)),g3)));
p3: ratsimp(second(divide(ev(y,second(l)),g2)));
p4: ratsimp(second(divide(ev(z,third(l)),g1)));

v: p4/g1+p3/g2+p2/g3+p1;


partfrac(f/g,x);

t1: -(%i-3)/(2*%i-2);
t2: (%i+3)/(-2*%i-2);
realpart(t1)+imagpart(t1)*%i;
realpart(t2)+imagpart(t2)*%i;






/*   267e   */

f: 2;
g:x*(x^2+2*x+2);
l0: solve(g=0);

g1: (x-realpart(ev(x,first(l0))) - imagpart(ev(x,first(l0)))*%i);
g2: (x-realpart(ev(x,second(l0))) - imagpart(ev(x,second(l0)))*%i);
g3: (x-realpart(ev(x,third(l0))) - imagpart(ev(x,third(l0)))*%i);


g: coeff(expand(g),x,hipow(expand(g),x))*g1*g2*g3;

ratsimp(g-x*(x^2+2*x+2));

p1: ratsimp(first(divide(f,g)));
f1: ratsimp(second(divide(f,g)));

l1: ratsimp(expand((a)*g1*g2 + (b)*g1*g3 + (c)*g2*g3 ));

A: matrix(
[coeff(coeff(l1,x,2),a),coeff(coeff(l1,x,2),b),coeff(coeff(l1,x,2),c),coeff(f1,x,2)],
[coeff(coeff(l1,x,1),a),coeff(coeff(l1,x,1),b),coeff(coeff(l1,x,1),c),coeff(f1,x,1)],
[coeff(coeff(l1,x,0),a),coeff(coeff(l1,x,0),b),coeff(coeff(l1,x,0),c),coeff(f1,x,0)]
);

echelon(A);

l: linsolve([
A[1,1]*x+A[1,2]*y+A[1,3]*z=A[1,4],
A[2,1]*x+A[2,2]*y+A[2,3]*z=A[2,4],
A[3,1]*x+A[3,2]*y+A[3,3]*z=A[3,4]
],[x,y,z]);


p2: ratsimp(second(divide(ev(x,first(l)),g3)));
p3: ratsimp(second(divide(ev(y,second(l)),g2)));
p4: ratsimp(second(divide(ev(z,third(l)),g1)));

v: p4/g1+p3/g2+p2/g3+p1;


partfrac(f/g,x);

t1: 1/(%i-1);
t2: (1)/(-%i-1);
realpart(t1)+imagpart(t1)*%i;
realpart(t2)+imagpart(t2)*%i;