6-1 Polynomial Functions

Objectives

Exploring Polynomial Functions

Modeling Data with a Polynomial Function

Vocabulary

A polynomial is a monomial or the sum of monomials.

The exponent of the variable in a term determines the degree of that term.

Ordering the terms by descending order by degree. This order demonstrates the standard form of a polynomial.

P(x) = 2x³ - 5x² - 2x + 5

Leading Coefficient

Cubic Term

Quadratic Term

Linear Term

Constant Term

Degrees of a Polynomial

Degree Name Using Degree

Polynomial Example Number of Terms

Name Using Number of Terms

0 Constant 6 1 Monomial

1 Linear x + 3 2 Binomial

2 Quadratic 3x²

3 Cubic 2x³ - 5x² - 2x 3 Trinomial

4 Quartic

5 Quintic 4Polynomial of 4

Terms

234 53 xxx

153 235 xxx

Write each polynomial in standard form. Then classify it by degree and by number of terms.

a. 9 + x3 b. x3 – 2x2 – 3x4

x3 + 9 –3x4 + x3 – 2x2

The polynomial is a quartic trinomial.

The term with the largest degree is x3,so the polynomial is degree 3.

It has two terms.The polynomial is a cubic binomial.

The term with the largest degree is –3x4, so the polynomial is degree 4. It has three terms.

Classifying Polynomials

x y0 2.82 54 66 5.58 4

Using a graphing calculator, determine whether a linear,

quadratic, or cubic model best fits the values in the table.

Enter the data. Use the LinReg, QuadReg, and CubicReg options of a graphing calculator to find the best-fitting model for each polynomial classification.

Graph each model and compare.

The quadratic model appears to best fit the given values.

Linear model Quadratic model Cubic model

Comparing Models

To estimate the number of employees for 1988, you can use the Table function option of a graphing calculator to find that ƒ(13) 62.72. According to the model, there were about 62 employees in 1988.

The table shows data on the number of employees that a small

company had from 1975 to 2000. Find a cubic function to model the data.

Use it to estimate the number of employees in 1998. Let 0 represent 1975.

To find a cubic model, use the CubicReg option of a graphing calculator.

The function ƒ(x) = 0.0096x3 – 0.375x2 + 3.541x + 58.96 is an approximate model for the cubic function.

1975 60

1980 65

1985 70

1990 60

1995 55

2000 64

Number ofEmployees

YearEnter the data.

Graph the model.

Real World Connection