Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 41. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Characterizing Polynomials 381 EXERCISE EXERCISE EXAMPLE SOLUTION In this course we will mainly be working with the real number system as opposed to the complex number system so unless otherwise stated when we ask about the number of roots it is understood that we mean real roots. Construct a polynomial equation with the specification given. The answers to this exercise are not unique a a third degree polynomial equation with roots at x -2 x 3 and x 0 b a second degree polynomial equation with a double root at x -1 c a second degree polynomial equation with no roots d a fifth degree polynomial equation with roots only at x -2 x 3 and x 0 Answers are provided at the end of the section. Identify which of the following two polynomials is possible to construct and construct it. a a fifth degree polynomial with no real zeros b a fourth degree polynomial with no real zeros Answer is provided at the end of the section. Finding the Zeros of a Polynomial Finding the zero of a linear function is simple. Finding the zeros of a quadratic is no problem when you use the quadratic formula. It is considerably harder to find the roots of a general cubic equation. A systematic procedure for finding a cubic s roots does exist but it is much more complicated than the quadratic formula and in practice is not often used. The story of the discovery of the cubic formula in Italy in the mid-1500s is one worth reading about in a math history book. It involves alleged lies about the discovery a public competition to solve cubic equations being won by Nicolo Tartaglia and his secret method being passed on to Girolamo Cardano in confidence but then published anyway and a final dispute from which one of the men is said to have been lucky to have escaped There is an algorithm for finding the zeros of fourth degree polynomials but in the early 1800s the Norwegian mathematician Niels Abel 1802-1829 proved that a formula for finding the zeros of a fifth degree polynomial
