Question

Consider $\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-c)(x-a)}{(b-c)(b-a)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=x^2,$ then which of the following is/are true

• A. Given equation is an identity
• B. $a,b,c$ are the roots of the given equation
• C. Given equation is not an identity
• D. Roots of the given equation are equal

Answer 1

Answer: (A), (B)

The correct options are
A Given equation is an identity
B $a,b,c$ are the roots of the given equation

$\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-c)(x-a)}{(b-c)(b-a)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=x^2$
By observation,
$x=a,x=b,x=c$ satisfy the given equation.
$\Rightarrow a,b,c$ are the roots
and there are $3$ roots which is greater than the degree of the given equation.
$\Rightarrow$ It is a identity.