Chapter 4: Polynomial and
Rational Functions
Section 4-1
Polynomial Functions
Polynomial in one variable
• A polynomial in one variable x is an expression of the form
a0 x n + a1 x n −1 + ....an x 0
• The coefficients a0 , a1 , a2 ...an represent complex
numbers (real or imaginary), a0 ≠ 0 ,and n represents a
nonnegative integer.
• The degree of a polynomial in one variable is the greatest
exponent of its variable. The coefficient of the variable with
the greatest exponent is called the leading coefficient.
18
10
5
• Example: For the expression
1000 x + 500 x + 250 x
18 is the degree and 1000 is the lead coefficient.
Polynomial function and zeros
•
If a function is defined by a polynomial in one variable with real coefficients, then
it is a polynomial function. If f (x) is a polynomial functions, the values of x for
which f (x) =0 are called the zeros of the function. (If the function is graphed,
these zeros are also the x-intercepts of the graph)
• f(x) = 3x4 - x3 + x2 + x – 1
•
Example: Consider the polynomial function
state the degree and the leading coefficient of the polynomial. Determine
whether -2 is a zero of f (x).
•
Solution: degree 4 and leading coefficient of 3.
•
Is -2 a zero?
•
No
3(-2)4-(-2)3+(-2)2+(-2)-1=0
48+8+4-2-1≠0
Classifying polynomials
• We can classify polynomials by their degree and also by their coefficients.
Complex Numbers
Our number system:
Real Numbers
Pure Imaginary
Numbers
Rational Numbers
Irrational numbers
integers
Fractions; repeating and terminating decimals
Negative integers
Whole numbers
Zero
Natural numbers
Difference between the term
zeros and roots
• The solution for a polynomial equation is
called a root. The words root and zero are
often used interchangeably, but technically,
you find the zero of a function and the root of
an equation.
Fundamental Theorem
of Algebra
• Every polynomial equation with degree
greater than zero has at least one root in the
set of complex numbers.
• The degree of a polynomial indicates the
number of possible roots of a polynomial
equation.
Graphs of polynomial functions
• On the next video you will see the graphs of
polynomial functions with various degrees. Pay
special attention to the pattern. These graphs are
also located on page 207 in your book.
• Also remember the graph of a polynomial function
with an odd degree MUST have at least one zero.
Find the polynomial equation
•
If you know the roots of a polynomial equation, you can use the corollary to the
fundamental theorem of Algebra to find the polynomial equation.
•
•
Example: Write a polynomial equation of least degree with roots 2, 3i,-3i.
Does the equation have an even or odd degree? How many times does the
graph of the related function cross the x axis?
•
Solution:
( x − 2)( x + 3i )( x − 3i ) = 0
( x − 2)( x 2 + 9) = 0
x 3 + 9 x − 2 x 2 − 18 = 0
x 3 − 2 x 2 + 9 x − 18 = 0
It is an odd degree. It crosses
the x axis once at 2. The other
two roots are imaginary.
Finding the roots
•
To find the roots of a polynomial equation factor it.
•
Example:
•
32x3 -32x2 + 4x -4 = 0
Since it is degree 3, there are
3 roots.
32 x 3 − 32 x 2 + 4 x − 4 = 0
Factor:
8x3 − 8x 2 + x − 1 = 0
(8 x 2 + 1)( x − 1) = 0
So (8 x 2 + 1) = 0 and (x - 1) = 0
i 2 i 2
which means roots are 1, ,
4
4
HW# 25
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Section 4-1
Pp. 210-212
#15-17,20,21,25-27,31-33,35
36,39,42,59,65