chap patterns of algorithms polynomial ab nowwe see what happens when we multiply the evaluations of and at points and - ab(- ( )( ab( ( )( ab( ( )( these results are the same as when we evaluate polynomial ab at these points note that evaluating any polynomial at is easy if we evaluate at and - we can share lot of the work between the two evaluations but we need five points to nail down polynomial ab fortunatelywe can speed processing for any pair of values and - this seems to indicate some promising ways to speed up the process of evaluating polynomials butevaluating two points in roughly the same time as evaluating one point only speeds the process by constant factor is there some way to generalized these observations to speed things up furtherand even if we do find way to evaluate many points quicklywe will also need to interpolate the five values to get the coefficients of ab back so we see that we could multiply two polynomials in less than th( operations if fast way could be found to do evaluation/interpolation of points before considering further how this might be donefirst observe again the relationship between evaluating polynomial at values and - in generalwe can write pa (xea (xoa (xwhere ea is the even powers and oa is the odd powers son/ - / - pa (xa + + = = the significance is that when evaluating the pair of values and -xwe get ea (xoa (xea (xoa (-xoa ( -oa (-xthuswe only need to compute the es and os once instead of twice to get both evaluations the key to fast polynomial multiplication is finding the right points to use for evaluation/interpolation to make the process efficient in particularwe want to take