The original question is not clear. There is a wide range of functionality to work with polynomials symbolically e.g. PolynomialReduce
, SymmetricReduction
, etc. however it is not clear why one just assumes that expr1
and expr2
are as given above and if they are unique representants of possible reductions in terms of third order polynomials of two variables, most likely they are not.
So I find that this question is asking: how can I find appropriate coefficients {a, b, c}
such that expr == p[x,y] + p[y,x]
where:
p[x_, y_] := a x^3 + b x^2 + c x y^2
There is SolveAlways
which helps in such cases:
{p[x, y], p[y, x]} /. SolveAlways[ expr == p[x, y] + p[y, x], {x, y}]
{{x^2 + 2 x^3 + 3 x y^2, 3 x^2 y + y^2 + 2 y^3}}
PolynomialReduce
demonstrates that expr1
and expr2
can express the original polynomial expr
PolynomialReduce[ expr, {expr1, expr2}, {x, y}]
{{1, 1}, 0}
SymmetricReduction
serves a different purpose, see e.g.
SymmetricReduction[expr, {x, y}, {expr1, expr2}]
Together[expr]
gives what you want. $\endgroup$expr1=expr2=expr/2
? The two add up to your original expression and can be obtained from each other by doingx
$\leftrightarrow$y
. The class of functions that does that is rather broad! How do you pick your target from within that? $\endgroup$