Consider the following bivariate polynomial $$ f(\lambda, R) = -6+4\lambda+6R-2R^2+2\lambda R^2-5\lambda R \, . $$
It can readily be seen that $f(2,1) = 0$. But, would it be possible to write $f(\lambda, R)$ as a product of two or more multivariate polynomials?
Thank you
If it were to factor as $g(\lambda,R)h(\lambda,R)$ with both $g,h$ non-constant then the degree of one of them in $\lambda$ would be $0$, since the degree of $f$ in $\lambda$ is $1$. Assume $h(\lambda,R)=h(R)$ is the one with degree $0$ in $\lambda$. Then there would be (complex) values $R_0$ of $R$ such that $h(R)=0$ (fundamental theorem of algebra). With this $R=R_0$ we have $f$ is $0$ for all values of $\lambda$. However, this is not the case since $f(\lambda,R)=0$ is equivalent to $$\lambda=\frac{6-6R+2R^2}{4+2R^2-5R}$$ So, $\lambda$ cannot be arbitrary to get $f(\lambda,R)=0$. Therefore, $h$ must be constant.