sage: var("x, y, z")
(x, y, z)
sage: Ex = (x-y)^2*(y-z)*(z-x) + (y-z)^2*(z-x)*(x-y) + (z-x)^2*(x-y)*(y-z)
sage: Ex
-(x - y)^2*(x - z)*(y - z) + (x - y)*(x - z)^2*(y - z) - (x - y)*(x - z)*(y -
z)^2
sage: Ex.expand()
0
sage: Ex.factor()
-(x - y)^2*(x - z)*(y - z) + (x - y)*(x - z)^2*(y - z) - (x - y)*(x - z)*(y -
z)^2
sage: Ex.simplify()
-(x - y)^2*(x - z)*(y - z) + (x - y)*(x - z)^2*(y - z) - (x - y)*(x - z)*(y -
z)^2
sage: Ex.simplify_full()
0
sage: Ex._sympy_().expand()._sage_()
0
sage: Ex._sympy_().factor()._sage_()
0
sage: Ex._giac_().factor()._sage_()
0
sage: Ex._fricas_().factor()._sage_()
0
sage: mathematica.Factor(Ex).sage()
0
So what ?
Sage’s factor doesn’t factorize this expression ? Big fail ! Film at 11…
As illustrated, Sage has lots of ways to factorize this expression. BTW,
expand a symbolic expression is often a clever way to (start to) finding a
“nice” expression of this quantity.
To understand why factorizing this expression is not trivial, meditate :
sage: Ex.operator()
<function add_vararg at 0x7ff7aaf70700>
sage: [u.expand() for u in Ex.operands()]
[-x^3*y + 2*x^2*y^2 - x*y^3 + x^3*z - x^2*y*z - x*y^2*z + y^3*z - x^2*z^2 +
2*x*y*z^2 - y^2*z^2,
x^3*y - x^2*y^2 - x^3*z - x^2*y*z + 2*x*y^2*z + 2*x^2*z^2 - x*y*z^2 - y^2*z^2
- x*z^3 + y*z^3,
-x^2*y^2 + x*y^3 + 2*x^2*y*z - x*y^2*z - y^3*z - x^2*z^2 - x*y*z^2 + 2*y^2*z^2
+ x*z^3 - y*z^3]
sage: [u.expand().cancel() for u in Ex._sympy_().args]
[x**3*y - x**3*z - x**2*y**2 - x**2*y*z + 2*x**2*z**2 + 2*x*y**2*z - x*y*z**2 -
x*z**3 - y**2*z**2 + y*z**3,
-x**3*y + x**3*z + 2*x**2*y**2 - x**2*y*z - x**2*z**2 - x*y**3 - x*y**2*z +
2*x*y*z**2 + y**3*z - y**2*z**2,
-x**2*y**2 + 2*x**2*y*z - x**2*z**2 + x*y**3 - x*y**2*z - x*y*z**2 + x*z**3 -
y**3*z + 2*y**2*z**2 - y*z**3]
sage: Ex._sympy_().cancel()
0
HTH,
Le dimanche 3 avril 2022 à 11:29:17 UTC+2, adarsh.k...@gmail.com a écrit :
> I checked this on my SageMath v9.6.beta5 on Ubuntu 20.04 LTS, and it
> checks out
>
> [image: Screenshot from 2022-04-03 14-58-46.png]
>
> On Sunday, April 3, 2022 at 2:57:35 PM UTC+5:30 Adarsh Kishore wrote:
>
>> By the way, f.expand() is not wrong. The expression is identically equal
>> to 0 for all values of x, y and z.
>>
>> https://www.mathpapa.com/algebra-calculator.html?q=(x-y)%5E2*(y-z)*(z-x)%2B(y-z)%5E2*(z-x)*(x-y)%2B(z-x)%5E2*(x-y)*(y-z)
>>
>> [image: Screenshot from 2022-04-03 14-55-22.png]
>>
>> As for `f.factor()`, the given expression cannot be factorized further.
>> So the answer is correct.
>> On Sunday, April 3, 2022 at 2:53:47 PM UTC+5:30 Adarsh Kishore wrote:
>>
>>> Which version of Sage are you using? And on which platform?
>>>
>>> On Sunday, April 3, 2022 at 10:44:59 AM UTC+5:30 a.simpl...@gmail.com
>>> wrote:
>>>
>>>> The following code
>>>>
>>>> > var("z y z")
>>>> > f = (x-y)^2*(y-z)*(z-x) + (y-z)^2*(z-x)*(x-y) + (z-x)^2*(x-y)*(y-z)
>>>> > f.expand()
>>>>
>>>> outputs 0.
>>>> But
>>>> > f.factor()
>>>> simply prints the original formula (x-y)^2...
>>>>
>>>> Maybe something is wrong?
>>>>
>>>
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