Polynomial functions

1. 7.1 Polynomial Functions

2. POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial
or a sum of monomials.
A POLYNOMIAL IN ONE
VARIABLE is a polynomial that
contains only one variable.
Example: 5x
2
+ 3x - 7

3. A polynomial function is a function of the form
f (x) = a
n
x
n
+ a
n – 1
x
n – 1
+· · ·+ a
1 x + a
0
Where a
n
¹ 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are
written in descending order of exponents from left to right.
For this polynomial function, a
n
is the leading coefficient,
a
0 is the constant term, and n is the degree.
a
n
¹ 0
a
n
a
n
leading coefficient
a
0
a
0
constant termn
n
degree
descending order of exponents from left to right.
n n – 1

4. POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.
What is the degree and leading
coefficient of 3x
5
– 3x + 2 ?

5. POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION .
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called
QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called
CUBIC POLYNOMIAL FUNCTIONS

6. Degree Type Standard Form
You are already familiar with some types of polynomial
functions. Here is a summary of common types of
polynomial functions.
4 Quarticf (x) = a
4
x
4
+ a
3
x
3
+ a
2
x
2
+ a
1
x + a
0
0 Constantf (x) = a
0
3 Cubic f (x) = a
3
x
3
+ a
2
x
2
+ a
1
x + a
0
2 Quadraticf (x) = a
2
x
2
+ a
1
x + a
0
1 Linear f (x) = a
1
x + a
0

7. Polynomial Functions
The largest exponent within the polynomial
determines the degree of the polynomial.
Quartic4
Cubic3
Quadratic2
Linear1
Name of
Function
Degree Polynomial
Function in General
Form
edxcxbxaxy ++++=
234
dcxbxaxy +++=
23
cbxaxy ++=
2
baxy +=

8. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f (x) = x
2

– 3x
4

– 7
1
2
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.
Its standard form is f (x) = – 3x
4

+ x
2
– 7.
1
2

9. Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
Identifying Polynomial Functions
The function is not a polynomial function because the
term 3
x
does not have a variable base and an exponent
that is a whole number.
SOLUTION
f (x) = x
3

+ 3
x

10. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
SOLUTION
f (x) = 6x
2

+ 2 x
–1
+ x
The function is not a polynomial function because the term
2x
–1
has an exponent that is not a whole number.

11. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 2, so it is a quadratic function.
The leading coefficient is p.
Its standard form is f (x) = p x
2
– 0.5x – 2.
f (x) = – 0.5 x + p x
2
– 2

12. f (x) = x
2

– 3 x
4

– 7
1
2
Identifying Polynomial Functions
f (x) = x
3

+ 3
x
f (x) = 6x
2

+ 2 x
–

1
+ x
Polynomial function?
f (x) = – 0.5x + p x
2
– 2

13. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(-2) if f(x) = 3x
2
– 2x – 6
f(-2) = 3(-2)
2
– 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10

14. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(2a) if f(x) = 3x
2
– 2x – 6
f(2a) = 3(2a)
2
– 2(2a) – 6
f(2a) = 12a
2
– 4a – 6

15. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(m + 2) if f(x) = 3x
2
– 2x – 6
f(m + 2) = 3(m + 2)
2
– 2(m + 2) – 6
f(m + 2) = 3(m
2
+ 4m + 4) – 2(m + 2) – 6
f(m + 2) = 3m
2
+ 12m + 12 – 2m – 4 – 6
f(m + 2) = 3m
2
+ 10m + 2

16. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find 2g(-2a) if g(x) = 3x
2
– 2x – 6
2g(-2a) = 2[3(-2a)
2
– 2(-2a) – 6]
2g(-2a) = 2[12a
2
+ 4a – 6]
2g(-2a) = 24a
2
+ 8a – 12

17. Examples of Polynomial Functions

18. Examples of Nonpolynomial Functions