# module 'poly' -- Polynomials

# A polynomial is represented by a list of coefficients, e.g.,
# [1, 10, 5] represents 1*x**0 + 10*x**1 + 5*x**2 (or 1 + 10x + 5x**2).
# There is no way to suppress internal zeros; trailing zeros are
# taken out by normalize().

def normalize(p): # Strip unnecessary zero coefficients
	n = len(p)
	while p:
		if p[n-1]: return p[:n]
		n = n-1
	return []

def plus(a, b):
	if len(a) < len(b): a, b = b, a # make sure a is the longest
	res = a[:] # make a copy
	for i in range(len(b)):
		res[i] = res[i] + b[i]
	return normalize(res)

def minus(a, b):
	if len(a) < len(b): a, b = b, a # make sure a is the longest
	res = a[:] # make a copy
	for i in range(len(b)):
		res[i] = res[i] - b[i]
	return normalize(res)

def one(power, coeff): # Representation of coeff * x**power
	res = []
	for i in range(power): res.append(0)
	return res + [coeff]

def times(a, b):
	res = []
	for i in range(len(a)):
		for j in range(len(b)):
			res = plus(res, one(i+j, a[i]*b[j]))
	return res

def power(a, n): # Raise polynomial a to the positive integral power n
	if n == 0: return [1]
	if n == 1: return a
	if n/2*2 == n:
		b = power(a, n/2)
		return times(b, b)
	return times(power(a, n-1), a)

def der(a): # First derivative
	res = a[1:]
	for i in range(len(res)):
		res[i] = res[i] * (i+1)
	return res

# Computing a primitive function would require rational arithmetic...