# Question:does Maple have method to factor on specific symbol only?

## Question:does Maple have method to factor on specific symbol only?

Maple 2021

I am looking for a robust way to factor an expression (if applicable) to become    x^n*(rest)  as we do it by hand.

edit The input will only be of type `+` and I am looking for a way extract a common factor to convert the input to term^n*(rest) where term is the common factor to pull out.

For example, given  x^2*Y+x and the symbol is given as x  the result will be x*(x*Y+1) and if the input is Y^2*x^3-x^3 then the output is x^3*(Y^2-1) and if there is no common factor x to pull out from all the terms, the output will be the same as the input.

I tried many commands and options, but can't find one method that works all the time for all cases.

For example for   x^2*Y+x  the command factor(x^2*Y+x) gives (Y - 1)*(Y + 1)*x^3 which is not what I want. There is no option to factor to give the name to factor on. And I did not know how to use the last argument of function to do that.

But here simplify(x^2*Y+x) happened to work on this and gives (Y^2 - 1)*x^3 but simplify does not work on first example. simplify(x^2*Y+x) returns the same expression back. So simplify is not reliable to use.

I tried collect, with options factor and simplify. Again, they work on one examples but not others.

collect(x^2*Y+x,x); does not do it. But collect(Y^2*x^3-x^3,x); works and returns (Y^2 - 1)*x^3 which is what I want.

The problem is that I do not know what the expression looks like. I just know the name and want to find if there is a common to any power that can be pulled out to rewite the expression as x^n*(rest) where is an integer or rational number depending.

This seems like a simple problem. But can't find a Maple command for.   I could ofcourse program it by brute force. Go over each term in the expression, check if each term has a free to any power in it multiplied by something else, then collect all these x^n term in a list. At end find the which is raised to lowest power, and then divide the whole expression by it.

Here is another way I can also try:  Use factor and also collect and also simplify. One at a time. Each time I check if the result is of type `*` but not a division! (check that denom is 1). If so, Then check if result has two operands only. If so, check if op(1,result) is for form x^anything. If so, then one of these cases worked.  Need to try this now to see if it will work on all cases I have.

Is there a better way to do this in Maple? It has to work on all expresions f(x) without knowing what the expression looks like.

update

I've updated the test cases and included all algorithms given to compare. It is hard in Maple to make a nice table to present results and keep math formatting below.

```restart;
makegrid := proc(M::Matrix)#https://www.mapleprimes.com/questions/202902-How-To-Create-Table-Like-Output-For
uses DocumentTools:-Layout;
local i,j,m,n,wks;
m,n := op(1,M);
wks := Worksheet(Table(alignment=center,width=20,
seq(Column(),j=1..n),
seq(Row(seq(Cell(Textfield(sprintf("%a",M[i,j]))),
j=1..n)),i=1..m)));
DocumentTools:-InsertContent(wks);
end proc:

acer_V1_common_factor := proc(x::algebraic, ee::algebraic)
local p, d := gcd(ee, x^frontend(degree,[ee,x]),'p');
d * p;
end proc:

acer_V2_common_factor := proc(x, ee) local d, t;
if ee::`+` then
t := max(map(proc(u) local r:=frontend(degree,[u,x]);
`if`(r::numeric,r,0); end proc,[op(ee)]));
d := gcd(numer(ee),x^t);
d*map(u->u/d,ee);
else ee; end if;
end proc:

dharr_common_factor:=proc(x,z)
local xn:=x^ldegree(collect(z,x),x);
if rem(z,xn,x)=0 then xn*quo(z,xn,x) else z end if;
end proc:

me_common_factor:=proc(term,expr)
local tmp;
local T1;

local update_T1:=proc()
T1:= hastype(op(1,tmp),identical(term)^anything) or hastype(op(1,tmp),identical(term));
if not T1 then
T1:= hastype(op(2,tmp),identical(term)^anything) or hastype(op(2,tmp),identical(term));
fi;
end proc;

if type(expr,`*`) or not has(expr,term) then
return expr;
fi;

tmp := collect(expr,term);
if type(tmp,`*`) and evalb(denom(tmp)=1) and evalb(nops(tmp)=2) then
update_T1();
if T1 then
return tmp;
fi;
fi;

tmp :=factor(expr);
if type(tmp,`*`) and evalb(denom(tmp)=1) and evalb(nops(tmp)=2) then
update_T1();
if T1 then
return tmp;
fi;
fi;

tmp := simplify(expr);
if type(tmp,`*`) and evalb(denom(tmp)=1) and evalb(nops(tmp)=2) then
update_T1();
if T1 then
return tmp;
fi;
fi;

return expr;
end proc:
############################

test_data:=[[x,x^2*Y+x],
[x,Y^2*x^3-x^3],
[x,x],
[x,x+2*x^2],
[x,x^4*diff(y(x),x)+x^7],
[x,x^4*diff(y(x),x)+x^7-sin(x)],
[y(x),y(x)^4*diff(y(x),x\$2)^2*diff(y(x),x)+y(x)^2*diff(y(x),x)+y(x)],
[y(x),y(x)^4*diff(y(x),x\$2)^2+y(x)^2*diff(y(x),x)+y(x)^9],
[x,x^4*y^2+x^2*y^2],
[y(x),y(x)^4*diff(y(x),x)^2+y(x)^2*diff(y(x),x)^2],
[diff(y(x),x),y(x)^4*diff(y(x),x)^2+y(x)^2*diff(y(x),x)^2],
[diff(y(x),x),y(x)*diff(y(x),x\$2)^2*diff(y(x),x)*sin(x)+diff(y(x),x)^3],
[y(x),diff(y(x),x)-(1+x^(1/2))/(1+y(x)^(1/2))],
[y(x),diff(y(x),x) -(x-1)*y(x)^5/x^2/(-y(x)+2*y(x)^3)],
[y(x),3*y(x)+diff(y(x),x) - 2*x/exp(3*x)],
[x,3*x^2*y^3+7*x/y],
[y,A-(1+x)/(1+y^(1/2))]
]:
RESULT:=Matrix(nops(test_data),6);
for N,item in test_data do
term:=item;
expr:=item;
RESULT[N,1]:=term; RESULT[N,2]:=expr;
try
result:=acer_V1_common_factor(term,expr);
if type(result,`*`) and denom(result)<>1 then
RESULT[N,3]:=expr;#bypass, not correct output
else
RESULT[N,3]:=result;#accept
fi;
catch:
RESULT[N,3]:=expr;#reject
end try;

try
result:=acer_V2_common_factor(term,expr);
if type(result,`*`) and denom(result)<>1 then
RESULT[N,4]:=expr;#bypass, not correct output
else
RESULT[N,4]:=result;#accept
fi;
catch:
RESULT[N,4]:=expr;#reject
end try;

try
result:=dharr_common_factor(term,expr);
if type(result,`*`) and denom(result)<>1 then
RESULT[N,5]:=expr;#bypass, not correct output
else
RESULT[N,5]:=result;#accept
fi;
catch:
RESULT[N,5]:=expr;#reject
end try;

try
result:=me_common_factor(term,expr);
if type(result,`*`) and denom(result)<>1 then
RESULT[N,6]:=expr;#bypass, not correct output
else
RESULT[N,6]:=result;#accept
fi;
catch:
RESULT[N,6]:=expr;#reject
end try;

od:

RESULT
```

how_to_do_special_factor.mw

Mapleprime will not let let insert content for some reason. Here is the output as screen shot but it is hard to read. But it is in the above worksheet.  ﻿