WEBVTT
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F G and Cure polynomial. Is that satisfied?
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This equation here for all exes except when Q A
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zero. And then we'd like to go ahead and
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show that F equals G for all ex. So
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here, I guess. Case. One. No
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way should consider, but it's kind of a silly
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case here. This is a Q is just the
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zero polynomial. So let me go ahead and put
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this bar here. So this is his cues.
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It's not like you've X's hero for some X.
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It's zero for every ex. So if this is
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true, then this we automatically have a people's G
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because we'LL never find an example in which F is
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not equal to G for some X satisfying. Q
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. Thanks not equal to zero because this can't happen
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because he was always zero. So in that case
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, it's vacuous. Lee Truth. There's really nothing
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to show then. The second case is Kyu is
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only zero for some X. So for finally,
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many exes, a polynomial could have on ly finally
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, many zeroes. So it's either always zero are
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on ly zero finally many times. So now if
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, for example, if Q of A is not
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equal to zero Then we can use this equation up
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here and we have enough of a over Cuba Guia
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que es. Since Cuba is non zero, we
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can just cancel these and then we see f of
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a equals jia come So a satisfies this equation here
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at those a values f equals gene. Now suppose
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that queue of a does equal zero This is what
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we'Ll have to use the continuity and that's what the
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hell is suggesting. So if ave this's weaken right
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, this is a limit as X approaches a f
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of x this's by continuity of f f is continuous
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because it's a polynomial This is given So then we
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can, right? This is the limit His ex
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approaches a of genomex So this is of course,
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when that satisfies this equation here can be zero in
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the denominator. No. So again, we have
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to make this assumption here. If we want this
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last equation now, using continuity of G, we
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can write this last expression Is she of eh using
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continuity of G. So this is because ji is
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continuous. But here's what we shown we shown for
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any hey whether queue of a zero or not.
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We always have equals G. So this shows that
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vehicle's G for all ex values, and that's your
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final answer.