1. Unit 3.3- Polynomial Equations Continued

2. Objectives  Divide polynomials with synthetic division  Combine graphical and algebraic methods to solve polynomial equations  Use the Fundamental Theorem of Algebra to find the number of complex solutions of a polynomial equation  Find the complex solutions of polynomial equations

3. Division of Polynomials; Synthetic Division Suppose f(x) is a cubic function and we know that a is a solution of f (x) = 0. We can write (x - a) as a factor of f (x), and if we divide this factor into the cubic function, the quotient will be a quadratic factor of f (x). If there are additional real solutions to f(x) = 0, this quadratic factor can be used to find the remaining solutions.

4. Long Division Synthetic Division

5. Example If x = 2 is a solution of x 3 + 6x 2 – x – 30 = 0, find the remaining solutions. Solution

6. Example (cont) Thus the quotient is ________________, with remainder 0, so x – 2 is a factor and The solutions are x = ___, __, and ___.

7. Multiplicity A graph touches but does not cross the x-axis at a zero of even multiplicity, and it crosses the x- axis at a zero of odd multiplicity.

8. Example The weekly profit for a product is P(x) = – 0.1x 3 + 11x 2 – 80x – 2000 thousand dollars, where x is the number of thousands of units produced and sold. To find the number of units that gives break-even, graph the function using a window representing up to 50 thousand units and find one x-intercept of the graph. We use the viewing window [0, 50] by [–2500, 8000].

9. Example (cont) b. Use synthetic division to find a quadratic factor of P(x). Solution

10. Example (cont) c. Find all of the zeros of P(x). Solution

13. Example Solve the equation 2x 4 + 10x 3 + 13x 2 – x – 6 = 0.