Putnam 1978 A2

Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$ . Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$ . Show that $$\text{det} \begin{pmatrix} p_1 & a& a & \cdots & a b & p_2 & a & \cdots & a b & b & p_3 & \cdots & a \vdots & \vdots & \vdots & \ddots & \vdots b & b& b &\cdots &p_n \end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$