Complex numbers and polynomials - University College Cork

Note especially that if a is rational and all the coeffs of p are rational, thenthe coeffs of q are also rational.If b is another root of p, then q(b) = 0, and so we can factor q: q(x) =(x − b)r(x), deg r = deg q − 1. Thus p(x) = (x − a)(x − b)r(x), and so on: ifx 1 , x 2 , . . . , x n are roots of p, thenp(x) = c(x − x 1 )(x − x 2 ) · · · (x − x n ) = cn∏(x − x i ).4. Suppose p/q, with p, q ∈ Z, (p, q) = 1, q ≠ 0, is a rational root of a polyp(x) = a n x n + · · · + a 0 whose coeffs are integers. Then p|a 0 , q|a n .This is another theorem due to Gauss. It reduces the search for rational rootsof polys with integer coeffs to an examination of a finite number of possibilitieswhich arise by factorising the integers a 0 , a n .5. How are the roots of a poly related to its coeffs? If ax 2 + bx + c is a quadratic,so that a ≠ 0, and its roots are x 1 , x 2 , thenax 2 + bx + c = a(x − x 1 )(x − x 2 ) = a(x 2 − (x 1 + x 2 )x + x 1 x 2 ,i=1whencex 1 + x 2 = − b a , x 1x 2 = c a .If ax 3 + bx 2 + cx + d is a cubic, so that a ≠ 0, and its roots are x 1 , x 2 , x 3 , thenax 3 + bx 2 + cx + d = a(x − x 1 )(x − x 2 )(x − x 3 )whence= a(x 3 − (x 1 + x 2 + x 3 )x + (x 1 x 2 + x 2 x 3 + x 3 x 1 )x − x 1 x 2 x 3 ),x 1 + x 2 + x 3 = − b a , x 1x 2 + x 2 x 3 + x 3 x 1 = c a , x 1x 2 x 3 = − d a .Similar relations hold for higher degree polynomials.6. The poly x n − 1 has roots ω k , k = 0, 1, 2, . . . , n whereThusso that, if x is a real number, thenω = cos( 2π n ) + i sin(2π n ) = e i2πn .n−1∏x n − 1 = (x − ω k ),k=0n−1∏|x n − 1| 2 = (x 2 − 2x cos( 2kπn ) + 1).k=019