Lesson 4-1

1. Lesson 4-1 Polynomial Functions

2. Vocabulary • Polynomial in one variable – an expression of the form . The coefficients represent complex numbers (real or imaginary) is not zero, and n represents a nonnegative integer. • Degree – the greatest exponent of a polynomials variable. • Leading coefficient – the coefficient of the variable with the greatest exponent. • Polynomial function – a function where P(x) is a polynomial in one variable.

3. Polynomial equation - a polynomial that is set equal to zero. • Root – a solution of the equation P(x) = 0 • Imaginary Number – a complex number of the form where and is the imaginary number. • Complex numbers – the imaginary numbers combined with the real numbers • Pure Imaginary Number – the complex number when a = 0 and • Fundamental Theorem of Algebra – Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers

4. Polynomial Functions

5. State the degree and the leading coefficient of the polynomial Degree of 4 and leading coefficient of 3 Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zero

6. State the degree and the leading coefficient of the polynomial Degree of 3 and leading coefficient of 1 Determine whether 6 is a zero of f(x) f(6) = 0 Yes it is a zero

7. State the degree and the leading coefficient of the polynomial Degree of 5 and leading coefficient of 8 Determine whether 1 is a zero of f(x) f(1) = 15 No

8. Example 2

9. Fundamental Theorem of Algebra

10. Corollary to the Fundamental Theorem of Algebra

14. Example 3

15. Write a polynomial equation of least degree with roots 2, 3i, and -3i

16. Write a polynomial equation of least degree with roots -5 and 7

17. Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i

18. Example 3

19. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis. Odd, once

20. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis. Even, twice

21. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis. Odd, once

22. State the number of complex roots of the equation has. Then find the roots and graph the related function.

23. The polynomial has a degree of 4 so there are 4 complex roots. 2 real roots and 2 imaginary roots for a total of 4 complex roots

24. Graph with the graphing Calculator

25. State the number of complex roots of the equation has. Then find the roots and graph the related function. 3 Complex Roots Graph on graphing calculator

26. Total of 3 Complex Roots,2 imaginary, 1 real x-1=0 x=1

27. State the number of complex roots of the equation has. Then find the roots and graph the related function. 2 Complex Roots Graph on graphing calculator

28. Total of 2 Complex Roots,2 real (double root) x-7=0 x=7

29. State the number of complex roots of the equation has. Then find the roots and graph the related function. 3 Complex Roots Graph on graphing calculator

30. 3 complex roots3 real roots x=0, x=-4, x=2

31. Example 5