6-3

1. 6-3 Polynomials Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

2. Warm Up Evaluate each expression for the given value of x. 2x+ 3; x = 22.x2+ 4; x = –3 3. –4x – 2; x = –1 4. 7x2 + 2x; x = 3 Identify the coefficient in each term. 5. 4x36. y3 7. 2n78. –s4

3. Objectives Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions.

4. VOCABULARY Monomial: a number, a variable, or a product of numbers and variables with whole-number exponents. The Degree of a monomial: the sum of the exponents of the variables (letters). *A constant (number only) has degree 0.

5. A. 4p4q3 Examples Together Find the degree of each monomial. The degree is 7. Add the exponents of the variables: 4 + 3 = 7. B. 7xy 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.

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

7. a. b. c. 1.5k2m 4x 2c3 You Try!!!! Find the degree of each monomial.

8. VOCABULARY (cont) Polynomial:a monomial or a sum or difference of monomials. The degree of a polynomial: the degree of the term with the greatest degree.

9. B. :degree 4 :degree 3 –5: degree 0 Example 2: Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11x7 + 3x3 11x7: degree 7 3x3: degree 3 Find the degree of each term. The degree of the polynomial is the greatest degree, 7. Find the degree of each term. The degree of the polynomial is the greatest degree, 4.

10. You Try!!!! Find the degree of each polynomial. a. 5x – 6 Find the degree of each term. b. x3y2 + x2y3 – x4 + 2 Find the degree of each term.

11. 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.

12. 6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9 2 Degree 1 5 2 5 1 0 0 –7x5 + 4x2 + 6x + 9. The leading The standard form is coefficient is –7. Example 3A: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in descending order:

13. y2 + y6 – 3y y6 + y2 – 3y Degree 6 6 1 2 1 2 The standard form is y6 + y2 – 3y. The leading coefficient is 1. Example 3B: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. y2 + y6 – 3y Find the degree of each term. Then arrange them in descending order:

14. Remember! A variable written without a coefficient has a coefficient of 1. y5 = 1y5

15. 16 – 4x2 + x5 + 9x3 x5 + 9x3 – 4x2 + 16 Degree 0 2 5 3 5 3 2 0 The leading x5 + 9x3 – 4x2 + 16. The standard form is coefficient is 1. Check It Out! Example 3a Write the polynomial in standard form. Then give the leading coefficient. 16 – 4x2 + x5 + 9x3 Find the degree of each term. Then arrange them in descending order:

16. 18y5 – 3y8 + 14y –3y8 + 18y5 + 14y Degree 8 1 5 8 5 1 The standard form is The leading –3y8 + 18y5 + 14y. coefficient is –3. Check It Out! Example 3b Write the polynomial in standard form. Then give the leading coefficient. 18y5 – 3y8 + 14y Find the degree of each term. Then arrange them in descending order:

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

18. 4y6 – 5y3 + 2y – 9 is a 6th-degree polynomial. Example 4: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5n3 + 4n 5n3 + 4n is acubic binomial. Degree 3 Terms 2 B. 4y6 – 5y3 + 2y – 9 Degree 6 Terms 4 C. –2x –2x is a linear monomial. Degree 1 Terms 1

19. 6 is a constant monomial. Check It Out! Example 4 Classify each polynomial according to its degree and number of terms. a. x3 + x2 – x + 2 x3 + x2 – x + 2 is acubic polynomial. Degree 3 Terms 4 b. 6 Degree 0 Terms 1 –3y8 + 18y5+ 14yis an 8th-degree trinomial. c. –3y8 + 18y5+ 14y Degree 8 Terms 3

20. –144 + 220 76 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 –16t2 + 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. –16t2 + 220 –16(3)2 + 220 The time is 3 seconds. –16(9) + 220 Evaluate the polynomial by using the order of operations.

21. 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 –16t2 + 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.

22. –400 + 2006 1606 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 –16t2 +400t + 6. How high will this firework be when it explodes? Substitute the time t to find the firework’s height. –16t2 + 400t + 6 –16(5)2 + 400(5) + 6 The time is 5 seconds. –16(25) + 400(5) + 6 Evaluate the polynomial by using the order of operations. –400 + 2000 + 6

23. 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 –16t2 +400t + 6. How high will this firework be when it explodes? When the firework explodes, it will be 1606 feet above the ground.

24. HOMEWORK PG. #