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1980 VTRMC #4

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August 11, 2020
algebrapolynomial

Problem Statement

Let P(x)P(x) be any polynomial of degree at most 3.3. It can be shown that there are numbers x1x_1 and x2x_2 such that ∫−11P(x) dx=P(x1)+P(x2),\textstyle\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2), where x1x_1 and x2x_2 are independent of the polynomial P.P.
(a) Show that x1=−x2.x_1=-x_2. (b) Find x1x_1 and x2.x_2.
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