MATH 274 Midterm: MATH274_BOYLE-M_SPRING2009_0101_MID_SOL_4

• School: University of Maryland
• Department: Mathematics
• Course: MATH 274
• Professor: All

Document Summary

Tuesday may 5, 2009: leibniz in 1702 claimed that for a real number a, the polynomial x4 + a2 could not be factored as a product of quadratic polynomials with real coe cients. Factor the poly- nomial x4 + 1 as a product of two such quadratics. Answer. x4 + 1 = (x2 + 2x + 1)(x2 . 2x + 1: let f (x) be the polynomial x3 + 6x2 + 7x + 1. Given a number c, let g(y) be the polyno- mial f (y + c). Find a number c such that g(y) has the form y3 + dy + e. explain, given a root r for g(y), how you can produce a root s for f (x). [this trick was cardano"s contribution to the general solution of the cubic: it reduced the general problem to the case where the coe cient of the quadratic term is zero.