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I am looking for "fast",pencil and paper technique for factoring a bi-variate quadratic polynomial,assume the polynomial is for the form $ax^2 + bxy + cy^2 + gx + fy + d$

where $a,b,c,g,f,d \in \mathbb{N_0}$.

Please explain with an example.

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    @Gerry Myerson:Sorry typo.Fixed now :)2011-10-14

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Let $y=0$, and factor the resulting one-variable quadratic (if possible) as (rx+s)(r'x+s'). Let $x=0$, and factor as (ty+u)(t'y+u'). Check to see whether your two factorizations are compatible. If so, they give you the factorization of the original; if not, there isn't one.

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    $f(x,y)=15x^2-7xy-2y^2$. $f(x,0)=15(x)(x)$, $f(0,y)=-2(y)(y)$. These are compatible, as all the constant terms are zero, so it's just a question of whether we can distribute the factors of the 15 and the -2. By systematically trying possibilities, we hit on $(3x-2y)(5x+y)$.2011-10-14