1. 4-1 Polynomial Functions

2. Objectives Determine roots of polynomial equations.
Apply the Fundamental Theorem of Algebra.

3. Polynomial in One Variable
a0, a1, a2, an: complex numbers (real or imaginary)
a0≠0
n: non-negative integer

4. Definitions
The degree of a polynomial is the greatest exponent of the variable.
The leading coefficient is the coefficient of the variable with the greatest exponent.
If all the coefficients are real numbers, it is a polynomial function.
The values of x where f(x)=0 are called the zeros (x-intercepts).

5. Example What is the degree? 4 What is the leading coefficient? 3
Is -2 a zero of f(x)? no

6. Imaginary Numbers Complex numbers: a+bi (a and b are real numbers)
Pure imaginary number: a=0 and b≠0

7. Imaginary Numbers

8. Fundamental Theorem of Algebra
Every polynomial equation with degree>0 has at least one root in the set of complex numbers
Corollary: A polynomial of degree n has exactly n complex roots.

9. Maximum Number of Roots
Degree: 1
Degree: 2
Degree: 3
Degree: 4
Degree: 5

10. Determining Roots
You can’t determine imaginary roots from the graph – you can only see the real roots.
Imaginary roots come in pairs.
An equation with odd degree must have a real root.

11. Finding the Polynomial
If you know the roots, you can find the polynomial.
(x-a)(x-b)=0

12. Example
Write a polynomial equation of least degree with roots 2, 3i and -3i.
(x-2)(x-3i)(x+3i)=0
(x-2)(x2-9i 2)=0
(x-2)(x2+9)=0
x3-2x2+9x-18=0
Does the equation have odd or even degree?
Odd
How many times does the graph cross the x-axis?
Once

13. Example
State the number of complex roots of the equation 32x3 - 32x2 + 4x - 4=0. Find the roots and graph.
32x3 - 32x2 + 4x - 4=0
32x2(x-1)+4(x-1)=0
(32x2+4)(x-1)=0
x2=-4/32
or x=1
x=±√-1/8
x=±i/2√2
=±i√2/4
Roots are 1 and ±i√2/4

14. Meterorology Example (#5)
A meteorologist sends a temperature probe on a small weather rocket through a cloud layer. The launch pad for the rocket is 2 feet off the ground. The height of the rocket after launching is modeled by the equation h=-16t2+232t+2 where h is the height of the rocket in feet and t is the elapsed time in seconds. When will the rocket be 114 feet above the ground?

15. Solution Find t when h=114. h=-16t2+232t+2 114=-16t2+232t+2
2t-1=0 or t-14=0
t=1/2 or t=14
(xscl=5, yscl=100)

16. Homework
page 210
#15-47 odds
Don’t graph 39-47