Download Define Polynomials Classifying Polynomials According to Their

Math 90
4.4 "Polynomials and Polynomial Functions"
Objectives:
*
De…ne and classify polynomials.
*
Evaluate and graph polynomial functions.
De…ne Polynomials
De…nition:
"Polynomials"
kA polynomial is an algebraic expression that is the sum of one or more terms containing whole number exponents on the variables.
5x + 3; 4n2
Examples:
6n
8; p3 + 3p2 q + 3pq 2 + q 3 ;
5
2 4
2 rs t
Classifying Polynomials According to Their Number of Terms
Polynomials with just one term are called
Polynomials with just two terms are called
Polynomials with just three terms are called
Example 1: (Classifying polynomials according to the number of terms)
Classify each polynomial as a monomial, binomial, trinomial, or none of these.
a) x2
c) 2x4
10x + 25
b)
7x3 + x2 + x
6x4
d) x5
6
5
Classifying Polynomials According to Their Degree
De…nition:
"Degree of a Term of a Polynomial"
The degree of a term of a polynomial in one variable is the value of the exponent on the variable. If a polynomial is in
more than one variable, the degree of a term is the sum of the exponents on the variables in that term. The degree of a
nonzero constant is 0 . The constant 0 has no de…ned degree.
De…nition:
"Degree of a Polynomial"
kThe degree of a polynomial is the same as the highest degree of any term of the polynomial.k
Polynomials According to Degree
Name
Degree
Linear
First-degree
Quadratic
Second-degree
Cubic
Third-degree
Example
Page: 1
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
4.4
Example 2: (Finding the degree of polynomials)
Find the degree of each polynomial.
a)
4x3
Term
5x2 + 3x
Coe¢ cient
b) 5x4 y 2 + 7xy 2 16x3 y 5
Term
Coe¢ cient
Degree
Degree
Evaluate Polynomial Functions
De…nition:
"Polynomial Functions"
kA polynomial function is a function whose equation is de…ned by a polynomial in one variable.k
Example 3: (Evaluating polynomials)
Evaluate the polynomial 3x2 + x
2 when x = 0, x =
2, and x = 3:
Functions and Function Notation
De…nition:
"Function"
Any equation in x and y where each value of x (the input) determines one value of y (the output) is called a function.
In this case, we say that y is a function of x.
The set of all input values x is called the domain of the function, and the set of all output values y is called the range.
Function Notation:
indicates that the variable y is a function of x:
The notation
Example 4: (Evaluating functions)
If y = f (x) = 4x2
a) f ( 2)
2x + 3, …nd each value.
b) f
Page: 2
1
2
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
4.4
Graphing Polynomial Functions
The Linear Function
Example 5: (Linear Function)
Graph f (x) = 2x
3
y
4
x
y
(x; y)
2
Domain :
-4
-2
2
4
x
-2
Range:
-4
The Square Function
Example 6: (Square Function)
Graph y = x2
y 10
x
y
(x; y)
8
6
Domain :
4
Range:
2
-6
-4
-2
2
4
6
-2
x
Cube Function
Example 7: (Cube function)
Graph y = x3
x
y
y
(x; y)
8
6
4
Domain :
2
Range:
-4
-2
-2
2
4
x
-4
-6
-8
Page: 3
Notes by Bibiana Lopez