the lesson to be learned from proposition is that if we perform an algorithm with nested loops such that the operations in the inner loop increase by one each timethen the total number of operations is quadratic in the number of timesnwe perform the outer loop to be fairthe number of operations is / / and so this is just over half the number of operations than an algorithm that uses operations each time the inner loop is performed but the order of growth is still quadratic in the cubic function and other polynomials continuing our discussion of functions that are powers of the inputwe consider the cubic functionf (nn which assigns to an input value the product of with itself three times this function appears less frequently in the context of algorithm analysis than the constantlinearand quadratic functions previously mentionedbut it does appear from time to time polynomials most of the functions we have listed so far can each be viewed as being part of larger class of functionsthe polynomials polynomial function has the formf (na ad nd where ad are constantscalled the coefficients of the polynomialand ad  integer dwhich indicates the highest power in the polynomialis called the degree of the polynomial for examplethe following functions are all polynomialsf ( ( ( (nn (nn thereforewe could argue that this book presents just four important functions used in algorithm analysisbut we will stick to saying that there are sevensince the constantlinearand quadratic functions are too important to be lumped in with other polynomials running times that are polynomials with small degree are generally better than polynomial running times with larger degree