Complex numbers and polynomials - University College Cork

6.3 Exercises1. Determine the roots of the cubic x 3 − (a 2 + ab + b 2 )x + ab(a + b).2. Suppose x 1 ≠ x 2 and y i = x 3 1, y 2 = x 3 2. Write down the equation of the linethrough the points (x 1 , y 1 ), (x 2 , y 2 ) and prove that it cuts the graph of y = x 3at three points, two of which may coincide.3. Prove that the cubic y = x 3 is convex on (0, ∞), and concave on (−∞, 0).7 AppendixThis contains additional information about polynomials and some IMO-type problems.7.1 Some more facts1. The collection of polys is closed under addition, multiplication and composition.In other words, ifp(x) = a 0 + a 1 x + · · · + a n x n , q(x) = b 0 + b x + · · · + b m x m ,are polys of degree n, m then p + q, pq, p ◦ q, q ◦ p are polys anddeg(p+q) ≤ max(deg p, deg q), deg(pq) = m+n, deg(p◦q) = deg(q ◦p) = mn.2. The following are special polynomials. Let a be a fixed complex number andn ∈ N:∑n−1n∑( nx n − a n , x n−k+1 a k , xk)n−k a k ,where( n=k)are the Binomial coefficients.Note thatk=0k=0{n!,(n−k)!k!if 0 ≤ k ≤ n,0, if k < 0 or k > n∑n−1x n − a n = (x − a) x n−k+1 a k , (x + a) n =k=0n∑k=0( nk)x n−k a k ,3. a is root (zero) of a poly p if p(a) = 0. If this is so, then we can factor p: thereis a poly q, with deg q = deg p − 1, such thatp(x) = (x − a)q(x).18