1. SOLVING SINGLE EQUATIONS Engineers are often required to solve complicated algebraic equations. These equations may represent cause-and-effect relationships between system variables, or they may be the result of applying a physical principle to a specific problem situation.

2. SOLVING SINGLE EQUATIONS

3. CHARACTERISTICS OF NONLINEAR ALGEBRAIC EQUATIONS A polynomial equation is a special case of a nonlinear equation. A polynomial equation can include powers of x but not terms such as log x, sin x, ex, etc. The following information is known about polynomial equations: 1. An nth-degree polynomial can have no more than n real roots. 2. If the degree of a polynomial is odd, there will always be at least one real root. 3. Complex roots always exist in pairs, if they exist at all. Each pair of complex roots consists of complex conjugates; that is, x 1 = u + iv and x 2 = u − iv.

4. Example 11.1 Identifying the Real Roots of a Polynomial

5. Problem 11.1 What can you conclude about the number of real roots for each of the following equations? (a) 3x + 10 = 0 (b) 3x 2 + 10 = 0 (c) 3x 3 + 10 = 0

6. Example 11.2 Solving a Polynomial Equation Graphically

7. Problem 11.4 Determine a real root for the equation given in Prob. 11.1(c) using the graphical method described above. (c) 3x 3 + 10 = 0

8. Problem 11.6