For comparison with other CAS choose from: Axiom Derive Macsyma Maple Reduce
Expand[ pol ]
pol := (a+b+c) ^4
4 3 2 2 3 4 3
a + 4 a b + 6 a b + 4 a b + b + 4 a c +
2 2 3 2 2 2
12 a b c + 12 a b c + 4 b c + 6 a c + 12 a b c +
2 2 3 3 4
6 b c + 4 a c + 4 b c + c
dpol = D[ Expand[pol], a ]
3 2 2 3 2
4 a + 12 a b + 12 a b + 4 b + 12 a c + 24 a b c +
2 2 2 3
12 b c + 12 a c + 12 b c + 4 c
D[ pol, a, b, 2 ]
24 (a + b + c)
Integrate[ dpol, a ]
4 3 2 2 3
a + 4 a (b + c) + 6 a (b + c) + 4 a (b + c)
% - pol
4 3 2 2 3
a + 4 a (b + c) + 6 a (b + c) + 4 a (b + c) -
4
(a + b + c)
Expand[%]
4 3 2 2 3 4
-b - 4 b c - 6 b c - 4 b c - c
(a ^2-b^2)/(a^2-2 a b+b^2)
2 2
a - b
---------------
2 2
a - 2 a b + b
Simplify[%]
a + b
-----
a - b
g := 34 x
^19 - 25 x^16 + 70 x^7 + 20 x^3 - 91 x - 86f1 := g (64 x
^34 - 21 x^47 - 126 x^8 - 46 x^5 - 16 x^60 - 81)f1
3 7 16 19
(-86 - 91 x + 20 x + 70 x - 25 x + 34 x )
5 8 34 47 60
(-81 - 46 x - 126 x + 64 x - 21 x - 16 x )
f1 = Expand[%]
3 5 6 7
6966 + 7371 x - 1620 x + 3956 x + 4186 x - 5670 x +
8 9 11 12 15
9916 x + 11466 x - 2520 x - 3220 x - 8820 x +
16 19 21 24 27
2025 x - 2754 x + 1150 x + 1586 x - 4284 x -
34 35 37 41 47
5504 x - 5824 x + 1280 x + 4480 x + 1806 x +
48 50 53 54 60
1911 x - 2020 x + 2176 x - 1470 x + 1376 x +
61 63 66 67 76
1456 x + 205 x - 714 x - 1120 x + 400 x -
79
544 x
f2 := g (72 x
^60 - 25 x^25 - 19 x^23 -22 x^39 - 83 x^52 + 54 x^10 + 81)f2 = Expand[f2]
3 7 10
-6966 - 7371 x + 1620 x + 5670 x - 4644 x -
11 13 16 17 19
4914 x + 1080 x - 2025 x + 3780 x + 2754 x +
23 24 25 26 28
1634 x + 1729 x + 2150 x + 545 x - 500 x +
29 30 32 39 40
1836 x - 1330 x - 1750 x + 2367 x + 2002 x +
41 42 44 46 52
625 x - 1086 x - 850 x - 1540 x + 7138 x +
53 55 58 59 60
7553 x - 1110 x - 748 x - 5810 x - 6192 x -
61 63 67 68 71
6552 x + 1440 x + 5040 x + 2075 x - 2822 x -
76 79
1800 x + 2448 x
PolynomialGCD[ f1, f2 ]
3 7 16 19
-86 - 91 x + 20 x + 70 x - 25 x + 34 x