Download polynomial function

Operations with Functions and
Polynomials
1-3 and 1-4
Unit 1
English Casbarro
Operations with Functions
You really already know how to do this. Recall, Combining Like Terms.
Ex. 4x4 - 5x2 + 6 – 7x4+ 10x2 -13.
You would combine: 4x4 -7x4 - 5x2 + 10x2 + 6 -13 = -3x4 + 5x2 - 7
Graphic Organizer-Function Notation
Fill in the following table.
Operation
Addition
Subtraction
Multiplication
Division
Notation
Warm-up: Function Notation
Given f(x) = 2x2 – 8 , g(x) = x2 + 5x + 6, and
h(x) = 2x + 4, find each function and
define the domain.
1. (f + g)(x)
2. (f – g)(x)
3. (g + h)(x)
4. (g – h)(x)
5. f(x) + h(x)
6. (fh)(x)
7.
f (x )
g (x )
8.
h (x )
f (x )
1-4: Polynomials
Definitions
A monomial is a number, a variable, or a product of both
A polynomial is a monomial or a sum or difference of monomials.
Each monomial in a polynomial is a term.
The degree of a monomial is the sum of the exponents of the variables.
The degree of a polynomial is the highest degree of all of the terms
of the polynomial.
The leading coefficient is the coefficient of the term with the highest degree.
A polynomial function is a function whose rule is a polynomial.
Identifying Polynomials:
Identifying Polynomials
Polynomials:
3x 4
Not Polynomials:
2z
3x
12
|2b3
+ 9z
– 6b|
½a
3
8
5y 2
7
0.15x
1
x
2
101
m0.75 – m
3t
2
– t
3
Ex. 1 Identifying the degree of a Monomial
Identify the degree of each monomial.
A. x4
B. 12
C. 4a2b
D. x 3y 4z
You Try:
Identify the degree of each monomial.
1a. x 3
1b. 7
1c. 5x 3y 2
1d. a 6bc 2
Standard Form of a Polynomial
Classifying by number of terms and by degrees.
Name
Monomial
Binomial
Trinomial
Terms
1
2
3
Example
7, x5, 3x
x5 - 7x
x2 + 2x + 3
Turn in the following problems:
1. Business The manager of a gift-basket business will ship the
baskets anywhere in the country. The cost to mail a basket
based on its weight, x, in pounds is given by
C(x) = 0.03x 3 – 0.75x 2 + 4.5x + 7.
a. What is the cost of shipping a 7-pound gift basket?
b. What is the cost of shipping a 19-pound gift basket?
2. Reasoning The total number of lights in a triangular lighting rig
is related to the triangular numbers, as shown below. The nth
triangular number is given by T (n )  1 n 2  1 n .
2
2
a. Write a polynomial function that represents the (n + 1)th
triangular number, T(n + 1).
b. The difference between two consecutive triangular numbers is
T(n + 1) – T(n). Subtract these two polynomial functions and state a
conclusion about the difference between consecutive triangular numbers.