Calculating P(2013) of Polynomial P(x) of Degree 2012

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The discussion focuses on calculating the value of the polynomial P(2013) for a polynomial P(x) of degree 2012, defined by the condition P(k) = 2^k for k = 0, 1, ..., 2012. Participants explore simpler polynomial cases, starting with degree 2 and degree 3, to identify patterns in polynomial coefficients and their corresponding outputs. The methodology involves using known values to derive coefficients and predict future outputs, ultimately leading to the conclusion that the pattern established can be generalized for higher degrees.

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P(x) is polynomial of degree 2012, P(k)=2^k, k=0,1,...,2012. Find P(2013)
 
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Suppose P is polynomial of degree 2, P(0)= 1, P(1)= 2, P(2)= 4. Since P is of degree 2 we can write it as [itex]P(x)= ax^2+ bx+ c[/itex]. Use P(0)= 1, P(1)= 2, P(2)= 4 to find a, b, and c. What is P(3)?

Suppose P(x) is a polynomial of degree 3, P(0)= 1, P(1)= 2, P(2)= 4, P(3)= 8. Since P is of degree 3, [itex]P(x)= ax^3+ bx^3+ cx+ d[/itex] use the values at x= 0, 1, 2, and 3 to find a, b, c, and d. What is P(4)?

Do you see a pattern? Can you show that this pattern is correct?
 

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