linear algebra - Determinant of block matrix as determinant of smaller matrix

I am considering block matrices $$\begin{pmatrix} A & v \\ v^T & x \end{pmatrix}$$ with $A \in \mathbb{R}^{(n-1) \times (n-1)}$, $v \in \mathbb{R}^{n-1}$, $x \in \mathbb{R}.$ Is there a rational $(n-1) \times (n-1)$ expression $p(A,v,x)$ I can form in the variables $A,v,x$ such that $$\mathrm{det}\begin{pmatrix} A & v \\ v^T & x \end{pmatrix} = \mathrm{det}(p(A,v,x))\; ?$$

The first thing I tried is the block matrix formula $$\mathrm{det}\begin{pmatrix} A & v \\ v^T & x \end{pmatrix} = x \, \mathrm{det}\Big(A - \frac{1}{x}vv^T \Big).$$ However writing this as just an $(n-1) \times (n-1)$ determinant introduces strange exponents: $$\mathrm{det}\Big( x^{1/(n-1)} A - x^{(2-n)/(n-1)} vv^T \Big)$$ which is obviously not polynomial unless $n = 2$. Since the result does not involve noninteger powers of $x$ I am hoping there is some expression $p$ that also only involves integer powers of $x$.

Answer

For blok matrices
$$
\left[
\begin{array}{c|c}
A_{11}& A_{12} \\

\hline
A_{21} & A_{22}
\end{array}
\right]
$$
if $A_{11}$ is invertible, we have the general formula ( you can see here or here):
$$
\det \left[
\begin{array}{c|c}
A_{11}& A_{12} \\

\hline
A_{21} & A_{22}
\end{array}
\right]=
\det A_{11}\cdot \det\left( A_{22}-A_{21}A_{11}^{-1}A_{12} \right)
$$
that, in your case becomes:
$$
\det \left[
\begin{array}{c|c}

A& v \\
\hline
v^T & x
\end{array}
\right]=
\det A\cdot \det\left(x-v^TA^{-1}v \right)=\left(x-v^TA^{-1}v \right)\det A
$$