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B.Math - interval

Source:

June 20, 2012
algebrapolynomialintegrationcalculusgeometrycalculus computations

Problem Statement

Let a0=0<a1<a2<...<ana_0=0<a_1<a_2<...<a_n be real numbers . Supppose p(t)p(t) is a real valued polynomial of degree nn such that
∫ajaj+1p(t) dt=0  ∀ 0≤j≤n−1\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1
Show that , for 0≤j≤n−10\le j\le n-1 , the polynomial p(t)p(t) has exactly one root in the interval (aj,aj+1) (a_j,a_{j+1})
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