How many solutions to the rational equation?

Question

If $a$ and $b$ and $c$ are parameters, how many solutions for:

$$\frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-a)(x-c)}{(b-a)(b-c)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1$$

I would say $3 \implies x = \{a, b, c\}$

But the answer is apparently $\infty$.

I am confused?

Answer 1

You observed that $a$, $b$, and $c$ are solutions. But if a polynomial of degree $\le 2$ has at least $3$ zeros, then it is the identically $0$ polynomial.

Answer 2

You are finding a second-degree polynomial whose graph goes through $(a,1),(b,1),(c,1)$ by Lagrange's interpolation. Such polynomial is unique, and it is clearly the constant $1$.