Tell Mathematica to factor something specific

Question

Is there a way to tell Mathematica to factor

$$ \frac{1}{1-x^2} $$

from a large expression?

For instance if the expression is A_1+A_2+A_3 and I want to factor 1/(1-x^2) from A_1+A_2 but leave A_3 alone.

Answer 1

Here is a very simple little function that takes two expressions k and p, and any function that accepts a single expression (like Expand, Together, Simplify,Apart,...). It then factors k from p and outputs a new expression $p=kq$. The parameter func_ operates on the form of $q$.

Here is the function:

sfactor[k_, p_, func_] := HoldForm[StandardForm[k]]*StandardForm[func@(p*1/k)]

The output is going to have unreleased hold forms in it thus limiting the usefulness of that output, but then again, I have no idea what you would want it for except to actually look at the factoring. Here are some examples:

Factor $x^2$ from $x^3+3x+1$:

sfactor[x^2, x^3 + 3 x + 1, Apart]

Output:

What it does to your abstract example:

sfactor[1/(1 - x^2), Subscript[a, 1] + Subscript[a, 2], Expand] + Subscript[a, 3]

Output:

And one more for giggles, here I know one of the binomial factors:

sfactor[x^2 + 3, 3 + 15 x + 22 x^2 + 11 x^3 + 7 x^4 + 2 x^5, Simplify]

Output:

Again, an expression involving unreleased hold forms would need to be fixed for further computations, but you know what it would simplify to in the end, the output always being $p$ expressed with the factor $k$.