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Given $(x,y)=(-1,3),(0,1),(1,-1),(-2,-1),(2,3)$. Show that both the Newton's divided-difference polynomial below, interpolate the data.

$P(x)=3-2(x+1)+(x+1)x(x-1)$ &
$Q(x)=-1+4(x+2)-3(x+2)(x+1)+(x+2)(x+1)x$

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By expanding and simplifying both $P(x)$ and $Q(x)$ can be seen to be equal to $x^3 - 3x + 1$. So clearly if either of them interpolate the data then both of them do.

As for checking that they interpolate the given points, you can just manually insert each of the $x$ values one by one and check that the functions indeed give an output equal to the given $y$ values.

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  • $\begingroup$ Okay, I got that, thanks! $\endgroup$ – Imtant Oct 28 '16 at 4:17

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