how to factor a^3b-ab^3+a^2+b^2+1

Question

 how to factor a^3b-ab^3+a^2+b^2+1?

Answer 1

The answer to this one is fairly simple, but the reasoning leading to the answer is not.

 

The factorization is

 

[ (a²+b²)/2 +ab- (a²-b²)/2 +1] [ (a²+b²)/2 -ab +(a²-b²) +1]

 

This can be checked by multiplying it all out.

 

The reasoning is that the form of the answer must be

 [ first quadratic expression +1 ] [ second quadratic expression +1]

The only quadratic forms that occur in the original expression are:

  a²+b² ,  ab , a² -b²

 

Thus the first and second quadratic expressions must be (different) linear combinations of these three.

So   first quadratic expression =  A(a²+b²) + B ab + C(a²-b²)  

and similarly for second quadratic expression with  A' , B' , C'

Multiplying everything out, and demanding that it equals the original expression

results in A = A' =1/2,  B = 1, B' = -1,  C = -1/2,  C' = 1/2