I actually don't know how to factor this expression.
$\displaystyle x^2-y^2+2yz-2zx-4x+2y+2z+3$
Hello, chengbin!
This is the Factoring Problem From Hell
. . coming soon to a theater near you.
Factor: .$\displaystyle x^2-y^2+2yz-2zx\:{\color{blue}-\:4x}\:{\color{red}+\:2y}+2z+3$
Replace $\displaystyle {\color{blue}-4x}$ with $\displaystyle {\color{blue}-x - 3x}$
Replace $\displaystyle {\color{red}2y}$ with $\displaystyle {\color{red}-y + 3y}$
We have: .$\displaystyle x^2 - y^2 + 2yz - 2zx \:{\color{blue}- x - 3x}\: {\color{red}- y + 3y} + 2z + 3$
Rearrange: .$\displaystyle x^2 - x - y^2 - y - 2zx + 2yz + 2z - 3x + 3y + 3 $
Subtract and add $\displaystyle {\color{red}xy}$:
. . $\displaystyle x^2 \:{\color{red}- \:xy} - x \:{\color{red}+\: xy} - y^2 - y - 2zx + 2yz + 2z - 3x + 3y + 3$
Factor: .$\displaystyle x(x-y-1) + y(x-y-1) - 2z(x-y-1) - 3(x-y-1) $
Factor: .$\displaystyle (x - y - 1)(x + y - 2z - 3)$ . . . . ta-DAA!
well, i don't think Kumon (whoever that is) would have a problem with this. maybe you should have said what you can do earlier. as we have no idea what is in this worksheet you speak of, we can only describe the conventional way to attack this problem. i don't see any easier way to factor this than our "adding zero" method
I'm going to kill that student...
It seems like he's trying to test me! This problem was an example!!!
I actually wasted some time digging up my old worksheets, then I found out.
Sorry for bothering you with this problem.
BTW, Soroban, your method of factoring is very smart.
In case you want to know how to do it without adding zero terms, and to prove that Kumon LOVES to torture students, here is the worksheet that I spent 20 minutes finding. There is also 20 pages (double sided) of worksheets with the same level of difficulty, exploring just about every method of factoring except adding zero terms. Factorization is probably the most outrageously hard topic taught in Kumon, along with maxima and minima.