\dfrac{a^2(x-b)(x-c)}{(a-b)(a-c)
} + \dfrac{b^2(x-c)(x-a)}{(b-c)(b-a)} + \dfrac{c^2(x-a)(x-b)}{(c-a)(c-b)} = x^2
Solution 2015/073)
This
can be done in 2 ways
method
1
expand
we
have 1st
term
=
\dfrac{a^2(x-b)(x-c)}{(a-b)(a-c)}
=
\dfrac{-a^2(x-b)(x-c)}{(a-b)(c-a)}
=\dfrac{-a^2(x-b)(x-c)(b-c)}{(a-b)(b-c)(c-a)}
=
\dfrac{x^2(- a^2(b-c)) +xa^2(b^2-c^2) -a^2bc(b-c)}{(a-b)(b-c)(c-a)}
similarly
we have 2nd
term
=
\dfrac{b^2(x-c)(x-a)}{(b-c)(b-a)}
=
\dfrac{x^2(- b^2(c-a)) +xb^2(c^2-a^2) -b^2ca(c-a)}{(a-b)(b-c)(c-a)}
3rd
term
=
\dfrac{c^2(x-a)(x-b)}{(c-a)(c-b)}
=
\dfrac{x^2(- c^2(a-b)) +x(c^2(a^2-b^2) -c^ab(a-b)}{(a-b)(b-c)(c-a)}
so
the sum =
\frac{(x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b)) +x(a^2(b^2-c^2) +
b^2(c^2-a^2) + c^2(a^2-b^2)) – (a^2bc(b-c) + b^2ca(c-a) +
c^2ab(a^2-b^2)}{(a-b)(b-c)(c-a))}
=
\dfrac{x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b)}{(a-b)(b-c)(c-a)}
numerator
= (x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b))
=
x^2(-a^2(b-c) – b^2c+ ab^2 -ac^2 +bc^2)
=
x^2(-a^2(b-c) – b^2c+ bc^2 -ac^2 +ab^2)
=
x^2(-a^2(b-c) – bc(b-c) + a(b^2-c^2))
=
x^2(-a^2(b-c) – bc(b-c) + a(b+c)(b-c))
=
x^2(b-c)(-a^2-bc+ab+ac)
=
x^2(b-c)(a-b)(c-a)
so
we have ratio = x^2
hence
proved
Aliter (Method 2)
This
can be proved as below
because
this is quadratic this cannot have more than 2 root.
But
x= a, x= b, x= c satisfy the condition.
Hence
this has to be identity.
Proved