Operations with Functions and Operations with Functions and PolynomialsPolynomials1-3 and 1-41-3 and 1-4

Unit 1Unit 1

English CasbarroEnglish Casbarro

Operations with FunctionsYou 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 Notation

Addition

Subtraction

Multiplication

Division

Warm-up: Function NotationGiven 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. 8. )()(

xgxf

)()(

xfxh

1-4: PolynomialsDefinitionsA 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 2z 12 + 9z 3 ½a 7 0.15x 101 3t 2 – t 3

Not Polynomials: 3x |2b3 – 6b| m0.75 – m 25

8y x

21

Ex. 1 Identifying the degree of a MonomialIdentify 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 Terms Example

Monomial 1 7, x5, 3x

Binomial 2 x5 - 7x

Trinomial 3 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 .

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.

nnnT21

21

)( 2