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Sounds like the Taylor polynomial with remainder, to me.
I can't think of a formal approach to this problem, but the $\displaystyle p(x)$ is nothing but the Lagrange polynomial interpolation for a set of n+1 data points, while the second $\displaystyle P(x)$ is the Taylor's series approximation of $\displaystyle f(x)$ about $\displaystyle x=x0$
Look up the definition of a nth order Taylor's polynomial approximation about a point x=xo and you will have both questions answered.
But it would be more useful if someone could throw light on the derivation of the Taylor series expansion.
1)http://en.wikipedia.org/wiki/Lagrange_polynomial
2)http://en.wikipedia.org/wiki/Taylor_series