 Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial "z""N" − 1, here into real-coefficient polynomials of the form "z""M" − 1 and "z"2"M" + "az""M" + 1.Another polynomial viewpoint is exploited by the Winograd FFT algorithm, which factorizes "z""N" − 1 into cyclotomic polynomials—these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors.Indeed, Winograd showed that the DFT can be computed with only O("N") irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.