6-3 Polynomials 6-3 Polynomials
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1 6-3 Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 1
2 Warm Up Evaluate each expression for the given value of x. 1.2x+ 3; x= x 2 + 4; x= x 2; x= x 2 + 2x; x= 3 69 Identify the coefficient in each term. 5.4x y n s 4 5
3 Objectives Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions.
4 Vocabulary monomial degree of a monomial polynomial degree of a polynomial standard form of a polynomial leading coefficient quadratic cubic binomial trinomial
5 A monomialis a number, a variable, or a product of numbers and variables with whole-number exponents. The degree of a monomialis the sum of the exponents of the variables. A constant has degree 0.
6 Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A.4p 4 q 3 The degree is 7. Add the exponents of the variables: = 7. B. 7ed The degree is 2. Add the exponents of the variables: 1+ 1 = 2. C. 3 The degree is 0. Add the exponents of the variables: 0 = 0.
7 Remember! The termsof an expression are the parts being added or subtracted. See Lesson 1-7.
8 Check It Out!Example 1 Find the degree of each monomial. a.1.5k 2 m The degree is 3. Add the exponents of the variables: = 3. b.4x The degree is 1. Add the exponents of the variables: 1 = 1. c. 2c 3 The degree is 3. Add the exponents of the variables: 3 = 3.
9 A polynomialis a monomial or a sum or difference of monomials. The degree of a polynomialis the degree of the term with the greatest degree.
10 Example 2: Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11x 7 + 3x 3 11x 7 : degree 7 3x 3 : degree 3 The degree of the polynomial is the greatest degree, 7. Find the degree of each term. B. :degree 3 :degree 4 5: degree 0 Find the degree of each term. The degree of the polynomial is the greatest degree, 4.
11 Check It Out!Example 2 Find the degree of each polynomial. a. 5x 6 5x: degree 1 6: degree 0 The degree of the polynomial is the greatest degree, 1. Find the degree of each term. b. x 3 y 2 + x 2 y 3 x x 3 y 2 : degree 5 x 2 y 3 : degree 5 x 4 : degree 4 2: degree 0 The degree of the polynomial is the greatest degree, 5. Find the degree of each term.
12 The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form. The standard form of a polynomialthat contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.
13 Example 3A: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6x 7x 5 + 4x Find the degree of each term. Then arrange them in descending order: 6x 7x 5 + 4x x 5 + 4x 2 + 6x + 9 Degree The standard form is coefficient is 7. 7x 5 + 4x 2 + 6x + 9. The leading
14 Example 3B: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. y 2 + y 6 3y Find the degree of each term. Then arrange them in descending order: y 2 + y 6 3y y 6 + y 2 3y Degree The standard form is coefficient is 1. y 6 + y 2 3y. The leading
15 Remember! A variable written without a coefficient has a coefficient of 1. y 5 = 1y 5
16 Check It Out!Example 3a Write the polynomial in standard form. Then give the leading coefficient. 16 4x 2 + x 5 + 9x 3 Find the degree of each term. Then arrange them in descending order: 16 4x 2 + x 5 + 9x 3 x 5 + 9x 3 4x Degree The standard form is coefficient is 1. x 5 + 9x 3 4x The leading
17 Check It Out!Example 3b Write the polynomial in standard form. Then give the leading coefficient. 18y 5 3y y Find the degree of each term. Then arrange them in descending order: 18y 5 3y y 3y y y Degree The standard form is coefficient is 3. 3y y y. The leading
18 Some polynomials have special names based on their degree and the number of terms they have. Degree Name Constant Linear Quadratic Cubic Quartic Quintic 6 or more 6th,7 th,degree and so on Terms or more Name Monomial Binomial Trinomial Polynomial
19 Example 4: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5n 3 + 4n Degree 3 Terms 2 B. 4y 6 5y 3 + 2y 9 Degree 6 Terms 4 C. 2x Degree 1 Terms 1 5n 3 + 4nis acubic binomial. 4y 6 5y 3 + 2y 9 is a 6 th -degreepolynomial. 2x is a linear monomial.
20 Check It Out!Example 4 Classify each polynomial according to its degree and number of terms. a. x 3 + x 2 x + 2 Degree 3 Terms 4 x 3 + x 2 x + 2 is a cubicpolynomial. b. 6 Degree 0 Terms 1 6 is a constantmonomial. c. 3y y y Degree 8 Terms 3 3y y yis an 8th-degree trinomial.
21 Example 5: Application A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial 16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds? Substitute the time for tto find the lip balm s height. 16t (3) The time is 3 seconds. 16(9) Evaluate the polynomial by using the order of operations.
22 Example 5: Application Continued A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial 16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds? After 3 seconds the lip balm will be 76 feet from the water.
23 Check It Out!Example 5 What if?another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by 16t t+ 6. How high will this firework be when it explodes? Substitute the time t to find the firework s height. 16t t+ 6 16(5) (5)+ 6 The time is 5 seconds. 16(25) + 400(5) Evaluate the polynomial by using the order of operations. 1606
24 Check It Out! Example 5 Continued What if?another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by 16t t+ 6. How high will this firework be when it explodes? When the firework explodes, it will be 1606 feet above the ground.
25 Lesson Quiz: Part I Find the degree of each polynomial. 1.7a 3 b 2 2a 4 + 4b x 2 3x 4 4 Write each polynomial in standard form. Then give the leading coefficient. 3.24g g 5 g 2 7g g 3 g ; x 4 + 3x 2 x 4 + 3x ; 1
26 Lesson Quiz: Part II Classify each polynomial according to its degree and number of terms x 2 12x+ 5 quadratic trinomial 6. 2x 4 1 quartic binomial 7. The polynomial 3.675v v 2 is used to estimate the stopping distance in feet for a car whose speed is vmiles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? ft
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2-2. Warm Up. Simplify each expression. 1. ( 7)(2.8) ( 9)( 9)
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The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.
Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number
Section 9.1: Add and Subtract Polynomials. The number part of a term with a variable part.
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17.1 Understanding Polynomial Expressions
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4.1 Understanding Polynomial Expressions
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Adding and Subtracting Polynomials Polynomials in the Sun
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