I have $P, Q, R: \ C \rightarrow \ C$ polynomials functions with maximum degree 2 and $a,b,c\in \ C$ such that $\begin{vmatrix} P(a) & Q(a) & R(a) \\ P(b) & Q(b) & R(b) \\ P(c) & Q(c) & R(c) \\ \end{vmatrix} = 1. $
I need to calculate $\begin{vmatrix} P(1) & Q(1) & R(1) \\ P(b) & Q(b) & R(b) \\ P(c) & Q(c) & R(c) \\ \end{vmatrix} +\begin{vmatrix} P(a) & Q(a) & R(a) \\ P(1) & Q(1) & R(1) \\ P(c) & Q(c) & R(c) \\ \end{vmatrix} + \begin{vmatrix} P(a) & Q(a) & R(a) \\ P(b) & Q(b) & R(b) \\ P(1) & Q(1) & R(1) \\ \end{vmatrix}$
I took $f(x)=\begin{vmatrix} P(x) & Q(x) & R(x) \\ P(b) & Q(b) & R(b) \\ P(c) & Q(c) & R(c) \\ \end{vmatrix}. $So $f(a)=1$ and $f(b)=f(c)=0$
How to continue?The result is $1$.
Let $\rho_x$ be the row vector with entries $P(x)$, $Q(x)$, and $R(x)$.
Let $A$ be the matrix with rows $\rho_a$, $\rho_b$, and $\rho_c$. Since $\det A\ne 0$, we know that $\rho_a,\rho_b,$ and $\rho_c$ form a basis for $\Bbb{R}^3$. Thus $\rho_x = C_a(x)\rho_a +C_b(x)\rho_b + C_c(x)\rho_c$ for some polynomials of degree at most two, $C_i\in \Bbb{R}[x]$, which we can find by multiplying $\rho_x$ by the inverse of $A$.
Observe that the sum of the determinants (where we replace $\rho_1$ in the determinants with $\rho_x$) is always $S(x)=C_a(x)+C_b(x)+C_c(x)$, by properties of determinants.
Then observe that $S(a)=1+0+0=1$, $S(b) = 0+1+0=1$, $S(c)=0+0+1=1$. Thus $S(x)$ is a quadratic polynomial which takes on the value 1 three times. Thus $S(x)$ must be identically 1.
In particular, $S(1)=1$.