MATHEMATICS Suppose a, b, c are three distinct real numbers. Let P(x)=\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)}When simplified, P(x) becomes

\frac{x^2+(a+b+c)(ab+bc+ca)}{(a-b)(b-c)(c-a)}

KVPY SA Exam 2011 Question Paper

Question : 1 of 80

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MATHEMATICS

Suppose a, b, c are three distinct real numbers. Let

P(x)={(x-b)(x-c)}/{(a-b)(a-c)}+{(x-c)(x-a)}/{(b-c)(b-a)}+{(x-a)(x-b)}/{(c-a)(c-b)}

When simplified, P(x) becomes

Solution:

P(x)={(x-b)(x-c)}/{(a-b)(a-c)}+{(x-c)(x-a)}/{(b-c)(b-a)}+{(x-a)(x-b)}/{(c-a)(c-b)}

Let P(a) = 1 + 0 + 0 = 1
P(b) = 0 + 1 + 0 = 1
P(c) = 0 + 0 + 1 = 1
∴ P(x) = 1 for all x ∈ R.

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