factor(x^4+1) and partfrac(1/(x^4+1))

10212019, 02:05 PM
(This post was last modified: 10212019 04:23 PM by rombva.)
Post: #1




factor(x^4+1) and partfrac(1/(x^4+1))
HP Prime G2 (SW Ver 2.1.14181, CAS Ver 1.4.9) can't factorize the expression x^4+1. It returns this expression unchanged.
Also, partfrac(1/(x^4+1)) can't return partial fractions. Is it a bug or a desired feature? 

10212019, 08:08 PM
Post: #2




RE: factor(x^4+1) and partfrac(1/(x^4+1))
x^4+1 has complex roots only. Go to the CAS settings and tick the setting "use i".
Then both commands will yield the factorization and partial factorization respectively. 

10222019, 01:48 AM
Post: #3




RE: factor(x^4+1) and partfrac(1/(x^4+1))
cfactor
cpartfrac There are specific commands for doing a complex factorization when not in complex mode. Generally, I recommend leaving the CAS settings as default and using the complex only versions only when you want. TW Although I work for HP, the views and opinions I post here are my own. 

10222019, 09:15 AM
Post: #4




RE: factor(x^4+1) and partfrac(1/(x^4+1))
I check out the tests from the document provided by link.
x^4+1=((x(1i)/sqrt(2))*(x(1+i)/sqrt(2)))*((x(1i)/sqrt(2))*(x(1+i)/sqrt(2)))= =(x^2sqrt(2)*x+1)*(x^2+sqrt(2)*x+1) I don't want to get a factorization result over the complex field. The polynomial x^4+1 may be rewriten as a product of polynomials over the field of reals (for example, HP 50g can factorize it). What's wrong with HP Prime's factorization in the real mode in this case? 

10222019, 02:57 PM
Post: #5




RE: factor(x^4+1) and partfrac(1/(x^4+1))
Factorisation is done on the smallest field containing the coefficients. Inside Xcas, you can explicitly add algebraic extensions of Q like this
partfrac(1/(x^4+1),sqrt(2)) This will probably be available in the HP Prime in a future firmware... 

10222019, 07:11 PM
Post: #6




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10212019 02:05 PM)rombva Wrote: HP Prime G2 (SW Ver 2.1.14181, CAS Ver 1.4.9) can't factorize the expression x^4+1. It returns this expression unchanged. From my experience with TI calcs, the TINspire CAS attempts to factor/expand any expression with linear, rational and real factors, the expression shown is not factorable based on these conditions: By specifying a variable, Nspire CAS is able to produce linear factors, as long as these factors contain only real numbers. Some polynomials are prime but, by specifying you want to factor in terms of x, you make sure Nspire produces linear factors (with approximations of irrational numbers). Using propfrac: I don't know if this applies to HP Prime, it's just a shot in the dark. Best, Aries 

10222019, 10:33 PM
(This post was last modified: 10232019 09:05 PM by Joe Horn.)
Post: #7




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10222019 09:15 AM)rombva Wrote: I don't want to get a factorization result over the complex field. The polynomial x^4+1 may be rewriten as a product of polynomials over the field of reals (for example, HP 50g can factorize it). What's wrong with HP Prime's factorization in the real mode in this case? Check your CAS settings. Near the bottom of the first page, in the left column, the second item is "Use square roots to factor polynomials". When that's turned on (the default setting), it yields the result you want: EDIT: Oops! You're right: The recent publicly released firmware versions return x^4+1 unchanged. The above screen shot was apparently from a different firmware version, possibly a beta version which didn't say "BETA" in the top corner. My mistake. Sorry! <0ΙΈ0> Joe 

10232019, 04:05 AM
Post: #8




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10222019 02:57 PM)parisse Wrote: Factorisation is done on the smallest field containing the coefficients. Inside Xcas, you can explicitly add algebraic extensions of Q like this How to find out the elements for algebraic extensions that provide factorization? 

10232019, 04:10 AM
Post: #9




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10222019 10:33 PM)Joe Horn Wrote: ... With default settings I get the input polynomial. 

10232019, 08:36 AM
Post: #10




RE: factor(x^4+1) and partfrac(1/(x^4+1))  
10232019, 11:45 AM
Post: #11




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10232019 08:36 AM)parisse Wrote:(10232019 04:05 AM)rombva Wrote: How to find out the elements for algebraic extensions that provide factorization? factor(x^4+1,lvar(cSolve(x^4+1=0,x))) works fine. For some polynomyals factor(f(x),lvar(cSolve(f(x)=0,x))) does nothing, for example, factor(x^7+2,lvar(cSolve(x^7+2=0,x))). For some polynomyals factor(f(x),lvar(cSolve(f(x)=0,x))) doesn't return complete factorization, for example, factor(x^7+1,lvar(cSolve(x^7+1=0,x))). For other polynomials the result contains a rootof([[][]]), for example, factor(x^5+2,lvar(cSolve(x^5+2=0,x))). How to find out the elements for algebraic extensions that provide factorization? How to get factorization for these cases? 

10232019, 12:08 PM
Post: #12




RE: factor(x^4+1) and partfrac(1/(x^4+1))
(10222019 10:33 PM)Joe Horn Wrote:(10222019 09:15 AM)rombva Wrote: I don't want to get a factorization result over the complex field. The polynomial x^4+1 may be rewriten as a product of polynomials over the field of reals (for example, HP 50g can factorize it). What's wrong with HP Prime's factorization in the real mode in this case? Hello Joe, can you please post your other settings? (CAS and Home) I get only the expression unchanged, like user rombva, no matter what settings I make. Many thanks in advance! 

10232019, 05:44 PM
Post: #13




RE: factor(x^4+1) and partfrac(1/(x^4+1))
I am in the same boat. When I put a decimal point after the 1, I get the desired approximation, but cannot get the exact result with a square root symbol...


10232019, 05:46 PM
Post: #14




RE: factor(x^4+1) and partfrac(1/(x^4+1))
I have all the choices in CAS settings on left checked, everything on the right unchecked. This on G2 2.0.0.13865 OS V2.060.650


10232019, 05:56 PM
Post: #15




RE: factor(x^4+1) and partfrac(1/(x^4+1))
to rombva: I was not clear enough. solve will help you find some algebraic extensions of Q (mainly sqrt()) but it will not help split every polynomial.
x^7+1 can be split with one rootof, factor(x^7+1,rootof(x^6x^5+x^4x^3+x^2x+1)) but that's not the generic case: in degree n, the extension may be of order n! and that would be too complicated even for small degrees. You always have the option to do numeric factorization, like this cfactor(x^7+1.0) (over C) or factor(x^7+1.0) (over R). 

10232019, 06:03 PM
Post: #16




RE: factor(x^4+1) and partfrac(1/(x^4+1))
Try this:
factor(x^4+1,sqrt(2)) with standard CAS settings road 

10232019, 06:03 PM
Post: #17




RE: factor(x^4+1) and partfrac(1/(x^4+1))  
10232019, 06:06 PM
Post: #18




RE: factor(x^4+1) and partfrac(1/(x^4+1))
and this:
cfactor(x^4+1,sqrt(2)) road 

10232019, 06:07 PM
Post: #19




RE: factor(x^4+1) and partfrac(1/(x^4+1))  
10232019, 07:15 PM
Post: #20




RE: factor(x^4+1) and partfrac(1/(x^4+1))
XCAS
road 

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