Let n € Z+ be a possitive integer, let Wo, W1, ..., Wn € C be distint complex nu
ID: 2945142 • Letter: L
Question
Let n € Z+ be a possitive integer, let Wo, W1, ..., Wn € C be distint complex numbers, and let Zo, Z1,..., Zn € C be any complex numbers. Then one can prove that there is a unique polynomial P(z) of degree at most n such that, for k € {0, 1,..., n}, P(Wk) = Zk.a. Find the unique polynomial of degree at most 2 that sastisfied P(0) = 0, P(1) =1, and P(2) = 2.
b. Can your result in part (a) be easily generalized to find the unique polynomial of degree at most n satisfying P(0) =0, p(1) = 1, ..., P(n) = n?
Explanation / Answer
a) p(z) = a_2z^2 + a_1z +a_o where a_o, a_1, a_2 are in C.
p(0) = 0 + 0 + a_o = 0
=> a_o = 0
p(1) = a_2 + a_1 + a_o = 1
a_2 + a_1 = 1
a_1 = 1 - a_2
p(2) = 4a_2 + 2a_1 + a_o = 2
4a_2 + 2(1 - a_2) = 2
2a_2 + (1 - a_2) = 1
a_2 + 1 = 1
a_2 = 0
a_1 = 1 - a_2
a_1 = 1
==> p(z) = z
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