7.1 An Introduction 7.1 An Introduction to Polynomialsto Polynomials

Objectives: Identify, evaluate, add, and Objectives: Identify, evaluate, add, and

subtract polynomials. Classify subtract polynomials. Classify polynomials, and describe the shapes of polynomials, and describe the shapes of

their graphs.their graphs.

Standard: 2.8.11.S. Analyze properties Standard: 2.8.11.S. Analyze properties and relationships of polynomials.and relationships of polynomials.

A monomial is a numeral, a variable, or the product of a numeral and one or

more variables.A monomial with no variables, such as –1

or , is called a constant.A coefficient is the numerical factor in a

monomial.The degree of a monomial is the sum of

the exponents of its variables.A polynomial is a monomial or a sum of

terms that are monomials.A polynomial with two terms is a

binomial.A polynomial with three terms is a

trinomial.The degree of a polynomial is the same

as that of its term with the greatest degree.

Ex 1. Classify each polynomial by degree and by number of terms.

First, combine Like terms with the same letter and exponent power.

Second, what is the highest exponent and name it by the degree.Third, count how many terms there are and name it.a.2x3 - 3x + 4x3 b. –2x3 + 3x4 + 2x3

+ 5

c. x2 + 4 - 8x - 2x3 d. 3x3 + 2 – x3 – 6x5

e. 5x + 2x3 + 4x2 f. x5 – 4x3 - x5 + 3x2 + 4x3

6x3 – 3x Cubic Binomial 3x4 + 5 Quartic Binomial

Cubic Polynomial 2x3 – 6x5 + 2 Quintic Trinomial

Cubic Trinomial 3x2 Quadratic Monomial

Adding and Subtracting Adding and Subtracting PolynomialsPolynomials

To add and subtract polynomials, To add and subtract polynomials, combine like terms.combine like terms.

The The standard formstandard form of a polynomial of a polynomial expression is written with the expression is written with the

exponents in exponents in descending orderdescending order of of degree.degree.

Ex 2. Find the sum and write it from Ex 2. Find the sum and write it from highest to lowest power.highest to lowest power.

a. (-2xa. (-2x22 – 3x – 3x33 + 5x + 4) + (-2x + 5x + 4) + (-2x33 + 7x – 6) + 7x – 6)

-5x-5x33 – 2x – 2x22 + 12x – 2 + 12x – 2

b. (2xb. (2x44 + 4x + 4x33 + 5x - 2) + (-2x + 5x - 2) + (-2x44 – 7x – 7x2 2 + 8x – + 8x – 10)10)

4x4x33 – 7x – 7x22 + 13x – 12 + 13x – 12

c. (6xc. (6x33 + 3x + 3x22 – 4) + (2x – 4) + (2x33 – 5x – 5x22 – 3x – 10) – 3x – 10)

8x8x33 – 2x – 2x22 – 3x – 14 – 3x – 14

Ex 3. Find the difference and write it Ex 3. Find the difference and write it

from highest to lowest power.from highest to lowest power. a. (-6xa. (-6x33 – 6x – 6x22 + 7x – 1) – ( 3x + 7x – 1) – ( 3x33 – 5x – 5x22 – 2x + – 2x +

8) 8)

-9x-9x3 3 – x– x2 2 + 9x – 9+ 9x – 9

b. ( 3xb. ( 3x33 – 12x – 12x22 – 5x + 1) – (-x – 5x + 1) – (-x22 + 5x + 8) + 5x + 8)

3x3x33 – 11x – 11x22 – 10x – 7 – 10x – 7

c. ( 5xc. ( 5x22 – 6x – 11) – (-8x – 6x – 11) – (-8x33 + x + x22 + 2) + 2)

8x8x33 + 4x + 4x22 – 6x – 13 – 6x – 13

Graphing Polynomial FunctionsGraphing Polynomial FunctionsA A polynomial functionpolynomial function is a function that is is a function that is

defined by a polynomial expression.defined by a polynomial expression.

Ex 4. Graph each function. Describe its Ex 4. Graph each function. Describe its general shape.general shape.

a.a. P(x) = 3xP(x) = 3x33 - 5x - 5x22 - 2x + 1 - 2x + 1 Cubic – S shapeCubic – S shapeb.b. Q(x) = xQ(x) = x44 - 8x - 8x22

Quartic – W ShapeQuartic – W Shapec.c. P(x) = -3x P(x) = -3x33 - 2x - 2x22 + 2x - 1 + 2x - 1 Cubic – S ShapeCubic – S Shape

Examine the shapes of the linear, Examine the shapes of the linear, quadratic, cubic, and quartic functions quadratic, cubic, and quartic functions

shown below.shown below.

Linear Quadratic Cubic Linear Quadratic Cubic QuarticQuartic

Line U S Line U S WW

Writing ActivitiesWriting Activities

1). Describe 2 different ways to classify 1). Describe 2 different ways to classify polynomials. Include examples.polynomials. Include examples.

2). What are the degree and the leading 2). What are the degree and the leading coefficient of Explain.coefficient of Explain.

3). Which of the following are like terms? 3). Which of the following are like terms?

Explain.Explain.

Review Of Introduction Review Of Introduction To PolynomialsTo Polynomials