1. Objectives: Students will be able to… Determine the number of zeros of a polynomial function Find ALL solutions to a polynomial function Write a polynomial function in standard form given the zeros 6.7: USING THE FUNDAMENTAL THEOREM OF ALGEBRA

2. If f(x) is a polynomial of degree n where n > 0, then the equation f(x)=0 has at least 1 root in the set of complex numbers. The number of solutions is related to the degree of the polynomial FUNDAMENTAL THEOREM OF ALGEBRA

3.  Example: x 2 – 6x + 9 = 0 (x-3) 2 = 0 x = 3 (repeated 2 times) We say “it has a multiplicity of 2”. IF A SOLUTION APPEARS TWICE, THEN IT IS A REPEATED SOLUTION

4.  In general, when all real and imaginary solutions are counted, an n th degree polynomial equation has exactly n solutions (and n zeros!!!)  Repeated solutions are counted individually Example: x 3 + 3x 2 +16x +48 =0 will have 3 solutions: -3, 4i, -4i RELATING THE DEGREE AND # OF SOLUTIONS

5. x 4 + 3x 3 – 8x 2 -22x -24 =0 HOW MANY SOLUTIONS DOES THE FOLLOWING POLYNOMIAL EQUATION HAVE? WHAT ARE THEY?

6. f(x) = x 3 + x 2 –x +15 How many x intercepts do you see on the graph? FIND ALL THE ZEROS.

7. f(x) = x 4 + 5x 2 -6 FIND ALL THE ZEROS

8.  Only real zeros appear as x intercepts  If (x – k) is a factor and raised to an even power, then the graph only touches the x axis at x = k  If (x – k) is a factor and raised to an odd power, then the graph crosses the x axis at x = k  Complex zeros always appear in conjugate pairs; if a + bi is a zero, so is a - bi FUN FACTS ABOUT ZEROS…

9. Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 1 and – 2 + i are zeros. EXAMPLE

10. Leading coefficient is 1; zeros are 5 and 2i EXAMPLE

11.  Enter function into Y =  Choose appropriate window  2 nd TRACE 2: zero  Move to the left and right of each zero as prompted Examples: Approximate the real zeros by graphing. 1.f(x) = x 3 – 4x 2 – 5x +14 2.f(x) = x 3 – 3x 2 – 2x +6 FINDING REAL ZEROS ON CALCULATOR

12. Graph f(x) = x 3 +2x 2 – 5x +1 on calculator. To find max and min values on calculator:  2 nd TRACE 3: Minimum or 4: Maximum  Follow prompts ****IMPORTANT: Y VALUE GIVES YOU MAX OR MIN VALUE, X VALUE TELLS YOU WHERE THE VALUE OCCURS FINDING MAXIMUM AND MINIMUM VALUES OF POLYNOMIAL FUNCTIONS

13. You want to make a rectangular box that is x cm high, (x +5) cm long, and (10- x) cm wide. What is the greatest possible volume? What would the dimensions of the box be? EXAMPLE

14. You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by cutting squares out of each corner and folding up the sides. You want the box to have the greatest possible volume. How much should you cut out of each corner? What is the max volume? What will the dimensions of the box be? EXAMPLE