6.1 Polynomial Functions At the end of this lesson, you should be able to:  classify polynomials  perform basic operations with polynomials  use polynomial operations to solve real-life problems

Vocabulary A monomial __________________________ _________________________________ __________________________________ A polynomial monomial or the sum of monomials___________________ is an expression that is either a real number, a variable, or the product of a real number and a variable

DEFINITION OF POLYNOMIAL IN X Polynomial functions are functions that can be written in this form: f ( x) = a n x n + a n-1 x n a 1 x + a 0

NOTEWORTHY: The value of n must be a nonnegative integer. The coefficients, as they are called, are an, a n-1,..., a 1, a 0. These are real numbers. The degree of the polynomial function is the highest value for n where a n is not equal to 0. The terms in the polynomial are shown in descending order by degree. This order illustrates the standard form of a polynomial. Polynomials with one, two, or three terms are called, monomials, binomials, and trinomials, respectively.

Standard form of a polynomial: P(x) = 2 x 3 -5x 2 - 2x + 5 Leading Coefficient Cubic Term Quad. Term Linear Term Constant Term

CLASSIFYING POLYNOMIALS Polynomials can be classified by _________ and by ______________. A polynomial of more then three terms has no special name. degreenumber of terms

DegreeName Using Degree Polynomial Example Number Of Terms Name by Terms 06 1x x 2 32x 3 – 5x 2 + 3x 4Quarticx Quintic-2x 5 + 6x 3 –x + 1 Complete the chart below:

ADDING, SUBTRACTING, AND MULTIPLYING POLYNOMIALS To add or subtract polynomials, we combine like terms (equivalent variables) using either horizontal or vertical format. Note: We will be using horizontal format.

(y 2 - 3y + 6) + (y - 3y 2 + y 3 ) (5x 2 + 2x +1) - ( 3x 2 – 4x –2) Adding and Subtracting Polynomials Examples:

Review: Special Product Patterns Sum and Difference (u + v) (u - v) = Example: (2x + 3) (2x – 3) = u 2 – v 2

Review: Special Product Patterns Square of a BinomialExample (u + v) 2 = (u - v) 2 = u 2 + 2uv + v 2 u 2 - 2uv + v 2

♥ Perfect Cubes: 1 3 = __2 3 = __3 3 = __4 3 = __5 3 = __6 3 = __ 7 3 = __8 3 = __9 3 = __10 3 = _20 3 = _30 3 = _ Note: Being familiar with perfect cubes will make quick mental math out of cubing a binomial!

Special Product Patterns Cube of a Binomial (u + v) 3 = Example: (x + 5) 3 = u 3 + 3u 2 v + 3uv 2 + v 3 (u - v) 3 = Example: (x - 3) 3 = u 3 - 3u 2 v + 3uv 2 - v 3 Learn-by- ♥ stuff!

CHECKING FOR UNDERSTANDING: Write each polynomial in standard form, then classify it by degree and number of terms. (x + 3) (x – 7) 9x 6 y 5 - 7x 4 y 3 + 3x 3 y x – 4 (2x + 5) 3 (2c – 3) (2c + 4) (c + 1) (2x + 5y) + (3x – 2y)(3x 3 + 3x 2 – 4x + 5) + (x 3 – 2x 2 + x – 4)

A POLYNOMIAL MODEL FOR VOLUME A rectangular box has sides whose lengths (in inches) are (2x + 1), (x + 2), and (x – 2). Write a polynomial, in standard form, for the volume of the box. Then find the volume of the box when x is 5 inches. x + 2 2x + 1 x - 2

MODELING DATA WITH A POLYNOMIAL FUNCTION x y Determine whether a linear, quadratic, or cubic model best fits the data, by using the LinReg, QuadReg, and CubicReg options of your graphing calculator to find the best-fitting model for each polynomial classifications.

Final Checks for Understanding 1. Perform the indicated operations, then classify the resulting polynomial by degree and number of terms: (3x 3 + 3x 2 – 4x + 5) + (x 3 – 2x 2 + x – 4) 2. Find the area of the blue region: 2x + 1 4x x x

HOMEWORK POLYNOMIAL FUNCIONS WS, PLUS TEXT PAGES ___________________