Linear Factors Mania

Algebra Level 3

Simplify the following expression, where $a$ , $b$ and $c$ are distinct real numbers. $\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$

1 0

\frac{3x^2}{(a-b)(b-c)(c-a)}

\frac{(x-a)(x-b)(x-c)}{(a-b)(b-c)(c-a)}

1 solution

Jaydee Lucero
Jun 1, 2014

Let $P(x)$ equals the expression. Then $P(a)=P(b)=P(c)=1$ . Also, $P$ is a function of at most second degree, for the numerators of each fraction in the expression are quadratic in form.

But a quadratic function can only take at most two distinct x-values to yield a single value. Hence, we conclude here that $P$ is not a quadratic function, but a linear function, and thus, a constant function. Thus, $P(x)=1$ for all real $x$ . Thus, the answer is $\boxed{1}$ .

My above solution is intuitive. You may provide a more technical solution for this problem. :)

Saan ka nag-aaral?

Kim Lehi Alterado - 6 years, 10 months ago

Sa UP Diliman po. :) Ikaw po?

Jaydee Lucero - 6 years, 10 months ago

Linear Factors Mania

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