Complex numbers and polynomials - University College Cork

The complex number ω, which depends on n, is never equal to 1, unless n = 1;it is called an nth root of unity. Its powers 1, ω, ω 2 , . . . , ω n−1 are also rootsof x n − 1, and form a cyclic group under multiplication. They are also thevertices of a regular n-gon inscribed in the unit circle. The perimeter of thisn-gon is given byn∑|ω k − ω k−1 | = n|1 − ω| = 2n sin( π n ),k=1which tends to 2π as n → ∞.7.2 Some identitiesIfp(x) = ∑ i≥0a i x i , q(x) = ∑ j≥0are two polys of degrees m, n, their product is also a poly of degree m + n, andp(x)q(x) = ∑ i≥0a i x i ∑ j≥0b j x jb j x j= ∑ a i b j x i+ji,j≥0= ∑ ∑a i b j x i+jk≥0 i+j=k= ∑ k≥0c k x k ,wherec k = ∑i+j=ka i b j = a 0 b k + a 1 b k−1 + · · · + a k b 0 , k = 0, 1, . . . , m + n.For instance, if p(x) = (1 + x) m , q(x) = ax 2 + bx + c, thenm∑( ) m(1 + x) m (ax 2 + bx + c) ≡ ( x i ) (b 0 + b 1 x + b 2 x 2 )i==i=0m+2∑x kk=0m+2∑k=0x k∑i+j=k,j≤2∑0≤j≤2( mi)b j( ) m+2m ∑b j = c k x k ,k − jk=0wherec 0 = b 0 = c, c 1 =( ) ( )m mb 0 + b 1 = mc + b,1 020