Polynomials lecture

A polynomial function has all of its variables with
exponents which are positive integers. It is not a
polynomial function if a variable has a negative
exponent or if the exponent is a fraction.
Polynomial Functions

A polynomial function is a function of the form:
  o
n
n
n
n a
x
a
x
a
x
a
x
f 



 
 1
1
1 
All of these coefficients are real numbers
n must be a positive integer
Remember integers are … –2, -1, 0, 1, 2 … (no decimals
or fractions) so positive integers would be 0, 1, 2 …
The degree of one variable polynomial is the largest
power on any x term in the polynomial.

A polynomial function is a function of the form
1
1 1 0
( ) n n
n n
f x a x a x a x a


    
where n is a nonnegative integer and each ai (i = 0, , n)
is a real number. The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each
polynomial function.
Polynomial Function Leading Coefficient Degree
5 3
( ) 2 3 5 1
f x x x x
    
3 2
( ) 6 7
f x x x x
   
( ) 14
f x 
-2 5
1 3
14 0

What is the degree of the following functions?

Which of the following are polynomial functions?
no
no
yes
yes

The Leading Coefficient
The polynomial function has a leading coefficient.
Once the function is written in descending order of
degree, the leading coefficient is the coefficient of the
term with the highest degree.

Find the leading coefficient and degree of each polynomial
function.
Polynomial Function Leading Coefficient Degree
5 3
( ) 2 3 5 1
f x x x x
    
3 2
( ) 6 7
f x x x x
   

Basic Features of Graphs of
Polynomial Functions.
• A graph of a polynomial function is
continuous. This means that the graph of a
polynomial function has no breaks, holes or
gaps.

NOT GRAPHS OF A POLYNOMIAL FUNCTION

Linear equations
General form: 𝑦 = 𝑚x + c
Example y= 2x + 1
y=2(-2)+1= -3 for x=-2
y=2(-1)+1= -1 for x=-1
y=2(0)+1= 1 for x=0
y=2(1)+1= 3 for x=1
y=2(2)+1= 5 for x=2
Now based on these coordinates we
can plot the graph as shown below.
Types of Polynomial function

Quadratic equations
General form Y= ax2+ 𝑏x + 𝑐 a≠0
Examples: Y = 5X2 + 2X + 3
Y = X2 +2
Y = X2
If leading Coefficient is Positive
If leading Coefficient is Negative

Quadratic equation in Chemistry

Cubic Function
General form Y= ax3+ 𝑏x2 + 𝑐x + d a≠0
Examples: Y = X3
X Y
0 0
1 1
2 8
3 27
-1 -1
-2 -8
-3 -27

END BEHAVIOR OF POLYNOMIAL FUNCTIONS
The behavior of the graph of a function to the far left and far right
is called its end behavior.
Although the graph of a polynomial function may have intervals
where it increases or decreases, the graph will eventually rise or
fall without bound as it moves far to the left or far to the right.
How can we determine the end behavior of a polynomial
function? We look only at the term with the highest degree.

The Leading Coefficient Test
Look for the term with the highest degree.
Is the coefficient greater than or less than 0?
Is the exponent even or odd?
The answers to these questions will help us to determine
the end behavior of the polynomial function.

If the leading coefficient is positive with an even
degree to its variable, the graph rises to the left and
rises to the right (, ).
Example: f(x) = x²

If the leading coefficient is negative with an even
degree to its variable, the graph falls to the left and
falls to the right (, ).
Example: f(x) = − x²

If the leading coefficient is positive with an odd
rises to the right (, ).
Example: f(x) = x³

If the leading coefficient is negative with an odd
degree to its variable, the graph rises to the left and
falls to the right (, ).
Example: f(x) = − x³

Using the Leading Coefficient Test
If the leading coefficient is positive with an
even degree to its variable, the graph rises to
the left and rises to the right (, ).

Determine the end behavior of the graph of…
f(x) = x³ + 3x − x − 3
If the leading coefficient is positive with an
odd degree to its variable, the graph falls to
the left and rises to the right (, ).

f(x) = − 2x³ + 3x − x − 3
If the leading coefficient is negative with an odd
degree to its variable, the graph rises to the left
and falls to the right (, ).

If the leading coefficient is negative with an
even degree to its variable, the graph falls to the
left and falls to the right (, ).

f(x) = 3x³(x − 1)(x + 5)
Because these terms and expressions are each
multiplied by each other, we add their degrees.
3 + 1 + 1 = 5
If the leading coefficient is positive with an odd
rises to the right (, ).

f(x) = − 4x³(x − 1)²(x + 5)
Add the degrees
If the leading coefficient is negative with an even
falls to the right (, ).

Zeros of Polynomial Functions
• It can be shown that for a polynomial function of
degree n, the following statements are true:
• 1. The function has, at most, n real zeros.
• 2. The graph has, at most, n – 1 turning points.
• Turning points (relative maximum or relative
minimum) are points at which the graph changes
from increasing to decreasing or vice versa.

Zeros of Polynomial Functions
The zeros of a polynomial function are the values of x which
make f(x) = 0. These values are the roots, or solutions of the
polynomial equation when y = 0. All real roots are the x-
intercepts of the graph.
How many turning points does f(x) = x³ + 3x² − x − 3 have?
How many Roots does f(x) = x³ + 3x² − x − 3 have? 3
But these may vary some times

Find all the real zeros of f (x) = x 4 – x3 – 2x2.
How many turning points are there?
The real zeros are x = –1, x = 0, and x = 2.
(where curve cross or touch X-axis )
These correspond to the
x-intercepts.
Turning points

Polynomials lecture

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