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4-1
Polynomial Functions
Objectives


Determine roots of polynomial
equations.
Apply the Fundamental Theorem of
Algebra.
Polynomial in One Variable
a0 x  a1 x
n
n 1
 ...an 2 x  an 1 x  an
2
a0, a1, a2, . . . an: complex numbers (real or
imaginary)
a0≠0
n: non-negative integer
Definitions
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


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).
Example
f ( x)  3 x  x  x  x  1
4
3
2
What is the degree?
4
What is the leading coefficient?
3
Is -2 a zero of f(x)?
f (2)  3(2)4  (2)3  (2)2  (2)  1  57
no
Imaginary Numbers
i  1
Complex numbers: a+bi (a and b are real numbers)
Pure imaginary number: a=0 and b≠0
Imaginary Numbers
i  1
i  1
2
i  i i  i
3
2
i  i i 1
4
2
2
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.
P( x)  k ( x  r1 )( x  r2 )...( x  rn )
Maximum Number of Roots
Degree: 1
Degree: 3
Degree: 2
Degree: 4
Degree: 5
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.
Finding the Polynomial

If you know the roots, you can find the
polynomial.

(x-a)(x-b)=0
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
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
x=±√-1/8
x=±i/2√2
=±i√2/4
or x=1
Roots are 1 and ±i√2/4
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?
Solution
Find t when h=114.
h=-16t2+232t+2
114=-16t2+232t+2
0=-16t2+232t-112
0=-8(2t2-29t+14)
0=-8(2t-1)(t-14)
2t-1=0 or t-14=0
t=1/2 or t=14
(xscl=5, yscl=100)
Homework
page 210
#15-47 odds
Don’t graph 39-47