Problem 3. Let f(T) be an integer polynomial. Its derivative f'(T) is defined to be the...

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Problem 3. Let f(T) be an integer polynomial. Its derivative f'(T) is defined to be the integer polynomial obtained from f(T) as follows: discard the constant term, then for each positive integer n, replace T" by nT-1 (here To means the constant 1). One can repeat this process to define what is the k-th derivative f(k) (T) of (T). (a) (5 pts) Give a formula of the degree of f(k) (T) in terms of the deg(f). Hint. First show that deg(f') = deg(f) - 1 as long as f 0. Be aware of what will happen when k > deg(f). (b) (5pts) Prove that taking derivative is compatible with modular reduction. Namely, if two integer polynomials f(T) and g(T) are congruence modulo m, then f'(T) and g'(T) are also congruence modulo m. Here m is any modulus. Problem 3. Let f(T) be an integer polynomial. Its derivative f'(T) is defined to be the integer polynomial obtained from f(T) as follows: discard the constant term, then for each positive integer n, replace T" by nT-1 (here To means the constant 1). One can repeat this process to define what is the k-th derivative f(k) (T) of (T). (a) (5 pts) Give a formula of the degree of f(k) (T) in terms of the deg(f). Hint. First show that deg(f') = deg(f) - 1 as long as f 0. Be aware of what will happen when k > deg(f). (b) (5pts) Prove that taking derivative is compatible with modular reduction. Namely, if two integer polynomials f(T) and g(T) are congruence modulo m, then f'(T) and g'(T) are also congruence modulo m. Here m is any modulus.