History of Mathematics: Homework 3

Homework Assignment 3

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

1. Examine the equation for real roots using Descartes Rule of signs, and also see if it has any rational roots

(a)

(b)

(c)

2. If 3 + 2i is the root of a quadratic (polynomial) equation with integer coefficients, find the quadratic equation.

3. Find a polynomial equation whose roots are:

(a) 2, 4, -5

(b) 3, 3, -1

(c) +i, -i, +i, -i and 12

4. Solve the following equations:

(a) 2x = 6

(b) 2x = 7

(c) 2x - 3(x - 4) = x + 11

(d)

(e)

(f)

(g)

(h)

Although equations were being solving in ancient times, the history of attempts to understand how to solve equations is quite fascinating. For a while, there used to be a separate undergraduate course entitled Theory of Equations that many colleges required undergraduate mathematics majors to take. This was true in the United States up until about 1960.

A relatively recent chapter in the history of solving equations has been the attempt to determine formulas for the solution of polynomial equations with integer coefficients in terms of the coefficients. For linear and quadratic equations such formulas are studied in high school. For cubic and quadratic equations such formulas are possible. However, we no know that for 5th degree equations and on, such formulas can not be found for all polynomial equations with integer coefficients.