1. Using Technology to Approximate Roots of Polynomial Equations

3. Let f denote a continuous function. If a<b And if f(a) and f(b) are of opposite sign, then the graph of f has at least one zero between a and b. Intermediate Value Theorem

4. To get a more precise answer, you can change the table setting.

6. Solving Polynomial Equations by Factoring Factoring by grouping Ex. 1. Solve:

7. Variable substitution Another method of solving higher-degree polynomial equations involves recognizing polynomials that have quadratic form. Ex. 2. Solve

8. The Rational Root Theorem Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term: If one of the roots of the equation P(x) = 0 is where P and q are nonzero integers with no common factor other than 1, then p must be a factor of and q must be a factor of

9. Ex. 3. a. According to the rational root theorem, what are the possible rational roots of is a possible rational root if p is a factor of –4 and q is a factor of 3.

10. b. Determine whether any of the possible rational roots really are roots. Then find all other roots, real or imaginary. Try synthetic substitution Or substitute in to see if any P(x) = 0 Use synthetic division

11. Find the other roots of Check x = -2 again because it may be a double root