7 6 Polynomials Warm Up Lesson Presentation Lesson

7 -6 Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Holt Mc. Dougal

7 -6 Polynomials Warm Up Evaluate each expression for the given value of x. 1. 2 x + 3; x = 2 7 3. – 4 x – 2; x = – 1 2 2. x 2 + 4; x = – 3 13 4. 7 x 2 + 2 x; x = 3 69 Identify the coefficient in each term. 5. 4 x 3 4 6. y 3 1 7. 2 n 7 2 8. –s 4 – 5 Holt Mc. Dougal Algebra 1

7 -6 Polynomials Objectives Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Vocabulary monomial degree of a monomial polynomial degree of a polynomial standard form of a polynomial leading coefficient Holt Mc. Dougal Algebra 1 quadratic cubic binomial trinomial

7 -6 Polynomials A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A. 4 p 4 q 3 The degree is 7. B. 7 ed The degree is 2. C. 3 The degree is 0. Holt Mc. Dougal Algebra 1 Add the exponents of the variables: 4 + 3 = 7. Add the exponents of the variables: 1+ 1 = 2. Add the exponents of the variables: 0 = 0.

7 -6 Polynomials Remember! The terms of an expression are the parts being added or subtracted. See Lesson 1 -7. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Check It Out! Example 1 Find the degree of each monomial. a. 1. 5 k 2 m The degree is 3. b. 4 x The degree is 1. c. 2 c 3 The degree is 3. Holt Mc. Dougal Algebra 1 Add the exponents of the variables: 2 + 1 = 3. Add the exponents of the variables: 1 = 1. Add the exponents of the variables: 3 = 3.

7 -6 Polynomials A polynomial is a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Example 2: Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11 x 7 + 3 x 3 11 x 7: degree 7 3 x 3: degree 3 The degree of the polynomial is the greatest degree, 7. Find the degree of each term. B. : degree 3 – 5: degree 0 : degree 4 Find the degree of each term. The degree of the polynomial is the greatest degree, 4. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Check It Out! Example 2 Find the degree of each polynomial. a. 5 x – 6 5 x: 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 4 + 2 x 3 y 2: degree 5 –x 4: degree 4 x 2 y 3: degree 5 2: degree 0 The degree of the polynomial is the greatest degree, 5. Holt Mc. Dougal Algebra 1 Find the degree of each term.

7 -6 Polynomials 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 polynomial that 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. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Example 3 A: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6 x – 7 x 5 + 4 x 2 + 9 Find the degree of each term. Then arrange them in descending order: 6 x – 7 x 5 + 4 x 2 + 9 Degree 1 5 2 0 – 7 x 5 + 4 x 2 + 6 x + 9 5 2 1 0 The standard form is – 7 x 5 + 4 x 2 + 6 x + 9. The leading coefficient is – 7. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Example 3 B: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. y 2 + y 6 – 3 y Find the degree of each term. Then arrange them in descending order: y 2 + y 6 – 3 y Degree 2 6 1 y 6 + y 2 – 3 y 6 2 1 The standard form is y 6 + y 2 – 3 y. The leading coefficient is 1. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Remember! A variable written without a coefficient has a coefficient of 1. y 5 = 1 y 5 Holt Mc. Dougal Algebra 1

7 -6 Polynomials Check It Out! Example 3 a Write the polynomial in standard form. Then give the leading coefficient. 16 – 4 x 2 + x 5 + 9 x 3 Find the degree of each term. Then arrange them in descending order: 16 – 4 x 2 + x 5 + 9 x 3 Degree 0 2 5 3 x 5 + 9 x 3 – 4 x 2 + 16 5 3 2 0 The standard form is x 5 + 9 x 3 – 4 x 2 + 16. The leading coefficient is 1. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Check It Out! Example 3 b Write the polynomial in standard form. Then give the leading coefficient. 18 y 5 – 3 y 8 + 14 y Find the degree of each term. Then arrange them in descending order: 18 y 5 – 3 y 8 + 14 y Degree 5 8 1 – 3 y 8 + 18 y 5 + 14 y 8 5 1 The standard form is – 3 y 8 + 18 y 5 + 14 y. The leading coefficient is – 3. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Some polynomials have special names based on their degree and the number of terms they have. Degree Name Terms Name 0 Constant 1 Monomial 1 Linear 2 Binomial 2 Quadratic Trinomial 3 4 Cubic Quartic 3 4 or more 5 Quintic 6 or more Holt Mc. Dougal Algebra 1 6 th, 7 th, degree and so on Polynomial

7 -6 Polynomials Example 4: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5 n 3 + 4 n Degree 3 Terms 2 5 n 3 + 4 n is a cubic binomial. B. 4 y 6 – 5 y 3 + 2 y – 9 Degree 6 Terms 4 4 y 6 – 5 y 3 + 2 y – 9 is a C. – 2 x Degree 1 Terms 1 – 2 x is a linear monomial. Holt Mc. Dougal Algebra 1 6 th-degree polynomial.

7 -6 Polynomials 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 cubic polynomial. b. 6 Degree 0 Terms 1 6 is a constant monomial. c. – 3 y 8 + 18 y 5 + 14 y Degree 8 Terms 3 – 3 y 8 + 18 y 5 + 14 y is an 8 th-degree trinomial. Holt Mc. Dougal Algebra 1

7 -6 Polynomials 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 – 16 t 2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? Substitute the time for t to find the lip balm’s height. – 16 t 2 + 220 – 16(3)2 + 220 – 16(9) + 220 – 144 + 220 76 Holt Mc. Dougal Algebra 1 The time is 3 seconds. Evaluate the polynomial by using the order of operations.

7 -6 Polynomials 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 – 16 t 2 + 220, 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. Holt Mc. Dougal Algebra 1

7 -6 Polynomials 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 – 16 t 2 +400 t + 6. How high will this firework be when it explodes? Substitute the time t to find the firework’s height. – 16 t 2 + 400 t + 6 – 16(5)2 + 400(5) + 6 The time is 5 seconds. – 16(25) + 400(5) + 6 – 400 + 2000 + 6 – 400 + 2006 1606 Holt Mc. Dougal Algebra 1 Evaluate the polynomial by using the order of operations.

7 -6 Polynomials 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 – 16 t 2 +400 t + 6. How high will this firework be when it explodes? When the firework explodes, it will be 1606 feet above the ground. Holt Mc. Dougal Algebra 1

7 -6 Polynomials Lesson Quiz: Part I Find the degree of each polynomial. 1. 7 a 3 b 2 – 2 a 4 + 4 b – 15 2. 25 x 2 – 3 x 4 5 4 Write each polynomial in standard form. Then give the leading coefficient. 3. 24 g 3 + 10 + 7 g 5 – g 2 7 g 5 + 24 g 3 – g 2 + 10; 7 4. 14 – x 4 + 3 x 2 Holt Mc. Dougal Algebra 1 –x 4 + 3 x 2 + 14; – 1

7 -6 Polynomials Lesson Quiz: Part II Classify each polynomial according to its degree and number of terms. 5. 18 x 2 – 12 x + 5 6. 2 x 4 – 1 quadratic trinomial quartic binomial 7. The polynomial 3. 675 v + 0. 096 v 2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727. 65 ft Holt Mc. Dougal Algebra 1