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

2. 7-6 Polynomials Additional Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A. 4p4q34p4q3 The degree is 7. Add the exponents of the variables: 4 + 3 = 7. B. 7ed The degree is 2. A variable written without an exponent has an exponent of 1. 1+ 1 = 2. C. 3 The degree is 0. There is no variable, but you can write 3 as 3x 0.

3. 7-6 Polynomials The terms of an expression are the parts being added or subtracted. See Lesson 1-7. Remember!

4. 7-6 Polynomials 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: 2 + 1 = 3. b. 4x4x The degree is 1. Add the exponents of the variables: 1 = 1. c. 2c32c3 The degree is 3. Add the exponents of the variables: 3 = 3.

5. 7-6 Polynomials Check up: p. 433 #’s 5,6

6. 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.

7. 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.

8. 7-6 Polynomials Write the polynomial in standard form. Then give the leading coefficient. Additional Example 2A: Writing Polynomials in Standard Form 6x – 7x 5 + 4x 2 + 9 Find the degree of each term. Then arrange them in descending order: 6x – 7x 5 + 4x 2 + 9 –7x 5 + 4x 2 + 6x + 9 Degree1 52 0 5 2 1 0 –7x 5 + 4x 2 + 6x + 9. The standard form is The leading coefficient is –7.

9. 7-6 Polynomials Write the polynomial in standard form. Then give the leading coefficient. Additional Example 2B: Writing Polynomials in Standard Form Find the degree of each term. Then arrange them in descending order: y 2 + y 6 − 3y y 2 + y 6 – 3y y 6 + y 2 – 3y Degree 2 6 12 16 The standard form is The leading coefficient is 1. y 6 + y 2 – 3y.

10. 7-6 Polynomials A variable written without a coefficient has a coefficient of 1. Remember! y 5 = 1y 5

11. 7-6 Polynomials Check It Out! Example 2a 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 2 + 16 Degree 02530235 The standard form is The leading coefficient is 1. x 5 + 9x 3 – 4x 2 + 16.

12. 7-6 Polynomials Check It Out! Example 2b Write the polynomial in standard form. Then give the leading coefficient. Find the degree of each term. Then arrange them in descending order: 18y 5 – 3y 8 + 14y –3y 8 + 18y 5 + 14y Degree 5 81 851 The standard form is The leading coefficient is –3. –3y 8 + 18y 5 + 14y.

13. 7-6 Polynomials Check up: p. 433 #’s 9,13

14. 7-6 Polynomials Some polynomials have special names based on their degree and the number of terms they have.

15. 7-6 Polynomials Classify each polynomial according to its degree and number of terms. Additional Example 3: Classifying Polynomials A. 5n 3 + 4n Degree 3 Terms 2 5n 3 + 4n is a cubic binomial. B. –2x Degree 1 Terms 1 – 2x is a linear monomial.

16. 7-6 Polynomials Classify each polynomial according to its degree and number of terms. Check It Out! Example 3 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. –3y 8 + 18y 5 + 14y Degree 8 Terms 3 –3y 8 + 18y 5 + 14y is an 8th-degree trinomial.

17. 7-6 Polynomials Check up: p. 433 #’s16,18

18. 7-6 Polynomials 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 2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? Additional Example 4: Application Substitute the time for t to find the lip balm ’ s height. – 16t 2 + 220 – 16(3) 2 + 200 The time is 3 seconds. – 16(9) + 200 Evaluate the polynomial by using the order of operations. – 144 + 200 76

19. 7-6 Polynomials 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 2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? Additional Example 5 Continued After 3 seconds the lip balm will be 76 feet above the water.

20. 7-6 Polynomials Check It Out! Example 4 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 2 + 400t + 6. How high will this firework be when it explodes? Substitute the time for t to find the firework ’ s height. –16t 2 + 400t + 6 –16(5) 2 + 400(5) + 6 The time is 5 seconds. –16(25) + 400(5) + 6 –400 + 2000 + 6 –400 + 2006 1606

21. 7-6 Polynomials Check It Out! Example 4 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 2 +400t + 6. How high will this firework be when it explodes? When the firework explodes, it will be 1606 feet above the ground.

22. 7-6 Polynomials Check up: p. 433 #’s 20

23. 7-6 Polynomials A root of a polynomial in one variable is a value of the variable for which the polynomial is equal to 0.

24. 7-6 Polynomials Additional Example 5: Identifying Roots of Polynomials Tell whether each number is a root of 3x 2 – 48. A. 4 3x 2 – 48 3(4) 2 – 48 3(16) – 48 48 – 48 4 is a root of 3x 2 – 48. 0 Substitute for x. Simplify. B. 0 3x 2 – 48 3(0) 2 – 48 3(0) – 48 0 – 48 – 48  0 is not a root of 3x 2 – 48.

25. 7-6 Polynomials Additional Example 5: Identifying Roots of Polynomials Tell whether each number is a root of 3x 2 – 48. C. – 4 3x 2 – 48 3( – 4) 2 – 48 3(16) – 48 48 – 48 – 4 is a root of 3x 2 – 48. 0 Substitute for x. Simplify.

26. 7-6 Polynomials Check It Out! Example 5 Tell whether 1 is a root of 3x 3 + x – 4. 3x 3 + x – 4 3(1) 3 + (1) – 4 3(1) + 1 – 4 3 + 1 – 4 1 is a root of 3x 3 + x – 4. 0 Substitute for x. Simplify.

27. 7-6 Polynomials Check up: p. 433 #’s 21, 23, 25