Question

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

Answer 1

Given equation is $\frac{(x-a)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}+\frac{(x-a)(x-b)}{(c-a)(c-b)}=1$ ........ $\left(i\right)$

As we know that if an integral function of $n$ dimensions vanishes for more than $n$ values of the variable, it must vanish for every value of the variable .

i.e. If an equation of $n$ dimensions has more than $n$ roots, then it is an identity.

Equation $\left(i\right)$ is of two dimensions and it is evidently satisfied by each of the three values $a,b$ and $c$.

Thus, it is an identity

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