CMPSC360hw6

# CMPSC360hw6 - CMPSC 360 Discrete Mathematics for Computer...

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Homework 6 Out: 2 Oct. Due: 9 Oct. Instructions: 1. [Polynomial interpolation] Consider the set of four points { (0 , 1) , (1 , 2) , (2 , 4) , (4 , 2) } . (a) Construct the unique degree-3 polynomial (over the reals) that passes through these four points by writing down and solving a system of linear equations. (b) Repeat part (a) but using the method of Lagrange interpolation. Show your working. 2. [Polynomials over GF ( p ) ] We have seen that a polynomial p ( x ) of degree d over a field F has at most d roots. However, it may have fewer than d roots. (a) Fermat’s Little Theorem states that, if p is prime, then for any a ∈ { 1 , 2 , . . . , p - 1 } , a p - 1 = 1 mod p . Deduce from this that any polynomial over GF ( p ) is equivalent to a polynomial of degree at most p - 1 . (b) Write down all polynomials over GF (2) that have no roots. Explain your answer. (c) Write down all polynomials over GF (3) that have no roots. Explain your answer. (d) Give an example of a polynomial that has at least one root over
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