# Factor completely with steps.   1-xy-20x^2y^2

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

I just realized that my terminology may not be completely standard, as I was trying to maintain an analogy between this and a problem with only one variable. Most people would say that the `-xy` term has degree 2 because that's the sum of the exponents, so "linear term" might not be the best choice. Still, the method itself is the most important part.

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

Just a reminder of the terminology:

The square term here is `-20x^2y^2,` the constant term is `1`, and the linear term is `-xy.`

I think the easiest way is to first find the product of the coefficient of the square term and the constant term, which is `-20*1=-20.` Now try to find two integers whose product is `-20` and whose sum is `-1` (because ```-1` is the coefficient of the `xy` term).

Those integers are `-5` and `4,` and that tells us that

`1-xy-20x^2y^2=(1-5xy)(1+4xy).`

The numbers `-5` and `4` become the coefficients of the `xy` terms in the factorization. This method can seem strange and hard to follow if you're not used to it, so if this is new for you it will take some practice for it to sink in. It can also be done by guess and check, just going through factors of `-20` and trying linear factors.