#Please do not post homework
#Caroline Cote, Jan 25th 2026, Assignment 1

#1
#for 1, i am going to copy and paste a few examples from the document i was working in and I will comment out the answers
evalf(sqrt(2));
#                          1.414213562

p := x^2 + 4*x + 4;
#                             2          
#                       p := x  + 4 x + 4

factor(p);
#                                   2
#                            (x + 2) 

rationalize(1/(1 + sqrt(2)));
#                            (1/2)    
#                           2      - 1
y := 'y';
#                             y := y

z := 'z';
#                             z := z

a := (x - y - z)*(x + y + z);
#                  a := (x - y - z) (x + y + z)

b := (x^2 - 2*x*y - 2*x*z + y^2 + 2*y*z + z^2)*(x^2 - x*y + x*z - y*z);
#     / 2                    2            2\ / 2                  \
#b := \x  - 2 x y - 2 x z + y  + 2 y z + z / \x  - x y + x z - y z/

c := a/b; normal(c);
#                           x + y + z             
#               ----------------------------------
#               / 2                  \            
#               \x  - x y + x z - y z/ (x - y - z)

simplify(c);
#                           x + y + z         
#                  ---------------------------
#                  (x + z) (x - y) (x - y - z)
factor(c);
#                           x + y + z         
#                  ---------------------------
#                  (x + z) (x - y) (x - y - z)

numer(c);
#                    (x - y - z) (x + y + z)

denom(c);
# / 2                    2            2\ / 2                  \
# \x  - 2 x y - 2 x z + y  + 2 y z + z / \x  - x y + x z - y z/

a := 2*x^3 + x^2 + 12;
#                              3    2     
#                      a := 2 x  + x  + 12
b := x^2 - 4;
#                                2    
#                          b := x  - 4

q := quo(a, b, x);
#                          q := 2 x + 1

r := rem(a, b, x);
#                         r := 8 x + 16

expand(a + (-b*q - r));
#                               0

p := (x + y + 1)*(x - y + 1)*(x - y - 1):
q := expand(%);
coeff(q, x, 2);
#                             -y + 1

degree(q, y);
#                               3

a := (x + y + z)*(x - y + z);
#                  a := (x + y + z) (x - y + z)

subs(x = 2, a);
#                    (2 + y + z) (2 - y + z)
eqn := x^2 - x = 1;
#                               2        
#                       eqn := x  - x = 1

R := solve(eqn, x);
#                     1   1  (1/2)  1   1  (1/2)
#                R := - + - 5     , - - - 5     
#                     2   2         2   2       

simplify(R[1]*R[2]);
#                               -1

lhs(eqn);
#                              2    
#                             x  - x

subs(x = R[1], lhs(eqn));
#                               2               
#                 /1   1  (1/2)\    1   1  (1/2)
#                 |- + - 5     |  - - - - 5     
#                 \2   2       /    2   2       

expand(%);
#                               1


eqn1 := x^3 + a*x = 14:
eqn2 := a^2 - x = 7:
solve({eqn1, eqn2}, {a, x});
#                 /
#                { 
#{a = 3, x = 2},  \
#
#            /  5       4        3        2              \  
#  a = RootOf\_Z  + 3 _Z  - 12 _Z  - 35 _Z  + 42 _Z + 119/, 
#
#                                                         2    \ 
#            /  5       4        3        2              \      }
#  x = RootOf\_Z  + 3 _Z  - 12 _Z  - 35 _Z  + 42 _Z + 119/  - 7/ 


%[1];
#                         {a = 3, x = 2}

subs(%, eqn1);
#                            14 = 14

polyeqn := Z^5 + 3*Z^4 - 12*Z^3 - 35*Z^2 + 42*Z + 119 = 0:
a1 := fsolve(polyeqn, Z);
#                       a1 := -3.136896207

x1 := a1^2 - 7;
#                       x1 := 2.840117813

subs({a = a1, x = x1}, {eqn1, eqn2});
#              {7.000000000 = 7, 14.00000002 = 14}

solve({a^2 - x = 7, x^3 + a*x = 14}, {a, x}):
%[1];
#                         {a = 3, x = 2}

2^(2^5) + 1;
#                           4294967297
ifactor(%);
#                        (641) (6700417)
x := 13/5:
floor(x);
frac(x);
floor(-x);
frac(-x);
#                               2
#
#                               3
#                               -
#                               5
#
#                               -3
#
#                               -3
#                               --
#                               5 
isprime(7);
#                              true

#2 and 3:
#done!

#4 and 5

Help:=proc(): print(`NuFP(pi),WtE(n,x)`):end:
with(combinat):

#part 4
#NuFP(pi)
NuFP := proc(pi) local i, a;
 
a := 0;
i:=1;
for i to nops(pi) do
   if pi[i] = i then
     a:= a+1;
   fi;
od;
a;
end:

#part 5
#WtE(n,x)
WtE := proc(n, x) local p, S;

if n=0 then
  1;
fi;

S:= 0;
for p in permute(n) do
  S := S + x^NuFP(p);
od;

expand(S);
end:

#k=0: [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
#A number: A000166

#k=1: [0, 1, 0, 3, 8, 45, 264, 1855, 14832]
#A number: A182390

#k=2: [0, 0, 1, 0, 6, 20, 135, 924, 7420]
#A number: A000387

#k=3: [0, 0, 0, 1, 0, 10, 40, 315, 2464]
#A number: no A number came up

#k=4: [0, 0, 0, 0, 1, 0, 15, 70, 630]
#A number: no A number came up

#k=5: [0, 0, 0, 0, 0, 1, 0, 21, 112]
#A number: no A number came up


#I am not attempting question 6.