The goals of big Oh-notation are to ignore constant factors which do not affect
exponential orders of growth, and to ignore the details which do not
impact the comparisons of algorithms. Abstract polynomial functions to
their highest degree term while ignoring all constants. 
becomes 
is UPPER bound on
such that for 
is LOWER bound
on 
such that for 
and 
such that for 
In studying the ASYMPTOTIC efficiency of algorithms, use only input sizes large enough to make only the order of growth of the running time relevant. The running time of an algorithm increases with the size of the input, as the size of the input increases without bound. As input size becomes extremely large, only the highest degree polynomial terms are of any significance, and even with ignoring constant factors, the growth rates themselves will provide a excellent indicator of whether a given algorithm will be able to run in a reasonable amount of time on a problem of a given size. Big Oh notation and worst-case analysis are tools that greatly simplify our ability to compare the efficiency of algorithms.