# Is it possible to write every real polynomial in two variables like ths one in this form?

Is it possible to write every real polynomial in two variables $$x$$ and $$y$$ with this form:

$$a x^2+b x y +c y^2$$

with general coefficients $$a,b,c$$ into the form

$$(d x + e y)^2$$

for some, possibly complex, $$d$$ and $$e$$?

From the second form to the first is obvious, but I need the other way around: is there some formula for the coefficients $$d$$ and $$e$$ as a function of $$a,b,c$$?

• How about $x^2+y^2$? Commented Jul 10, 2020 at 14:50

No, this does not happen for most polynomials. You can just divide the first expression thru by $$y^2$$ and obtain the expression $$a \left(\frac{x}{y}\right)^2 + b \left(\frac{x}{y}\right) + c$$ which is just a single-variable quadratic in $$u = x/y$$. It is well-known that the Fundamental Theorem of Algebra guarantees a factorization of this quadratic of the form $$a (u - r_1) (u - r_2)$$ but the roots can be different, so it may not be a perfect square. To recover the factorization for your original expression, just multiply thru by $$y^2$$ and you will see that it factors into $$a(x - r_1 y)(x - r_2 y) = (x\sqrt{a} - r_1 y\sqrt{a})(x\sqrt{a} - r_2 y\sqrt{a})$$ And hence you can see that such a factorization exists if and only if $$r_1 = r_2$$, which is equivalent to asserting that the discriminant $$b^2 - 4ac = 0$$.
No it is not possible, $$a = d^2$$ and $$c = e^2$$. If this is true, then $$d$$ and $$e$$ must be real because a complex number of the form $$t+ui$$ where $$u \neq 0$$ is always going to be complex (verify this yourself). Thus, there are three constraints on two variables meaning that not every real polynomial will be spanned by this form even if $$d$$ and $$e$$ can be complex (because if they have imaginary parts the polynomial itself wont be a real polynomial anymore).