# Factorization of a polynomial of degree 2

I want to find a factorization of the following polynomial of degree 2.

Let $$\alpha >0, \beta > 0,$$ and $$\gamma \geq 1$$. Is there any hope to factorize the following polynomial: $$\alpha^2 x^2 + \beta^2 y^2 - 2 \alpha \beta \gamma xy = C(Ax- By)^2,$$ for some positive constants $$A, B, C$$.

S

• I just edited your post. The word is 'polynomial' in English (in fact, you spelt it correctly in your first line!). Nov 15, 2019 at 10:52
• Well done. Thank you! Nov 15, 2019 at 10:55

If we expand the RHS first:

$$C(Ax-By)^2=C(A^2x^2+B^2y^2-2ABxy)=CA^2x^2+CB^2y^2-2ABCxy$$

Comparing coefficients with the LHS now:

$$CA^2=\alpha^2$$ (1)

$$CB^2=\beta^2$$ (2)

$$-2ABC=-2\alpha\beta\gamma\implies ABC=\alpha\beta\gamma$$

Multiplying together (1) and (2), we have $$C^2A^2B^2=\alpha^2\beta^2$$, which means that your polynomial can only be factored into the form $$C(Ax-By)^2$$ if $$\gamma=1$$.

• Even more, I think it an "iff" $\gamma = 1$. Thank you for your suggestions. Actually I try to lower bound this polynomial. Please could you check this math.stackexchange.com/questions/3436633/… Nov 15, 2019 at 11:15

Your expression can be factorized of we set $$\gamma = \frac{1}{\sqrt{\alpha\beta}}$$. In that case, $$C=1, A=\sqrt{\alpha}$$, and $$B=\sqrt{\beta}$$.
• Sorry I edit the polynom to $\alpha^2 x^2 + \beta^2 y^2 - 2 \alpha \beta \gamma xy$. When I did some identification using development I get the following: $$A = \frac{\alpha \gamma}{\beta}, B = \gamma, C = \frac{\beta^2}{\gamma^2}$$ However developing $C(Ax- By)^2 = \alpha^2 x^2 + \beta^2 y^2 - 2\alpha\beta xy$ (there is no $\gamma$ term in front of the monom $xy$. :( Nov 15, 2019 at 10:30