Warm Up 4-28-08 Evaluate each expression for the given value of x .

1. Warm Up 4-28-08 Evaluate each expression for the given value of x. 1. 2x + 3; x = 22.x2+ 4; x = –3 3. –4x – 2; x = –1 4. 7x2 + 2x = 3 Identify the coefficient in each term. 5. 4x36. y3 7. 2n78. –54 7 13 2 69 4 1 –1 2

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

3. Vocabulary monomial degree of a monomial polynomial degree of a polynomial standard form of a polynomial leading coefficient quadratic cubic binomial trinomial

4. 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 exponentsof the variables. A constant has degree 0.

5. Directions: Find the degree of each monomial.

6. A. 4p4q3 Example 1 The degree is 7. Add the exponents of the variables: 4 + 3 = 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 terms of an expression are the parts being added or subtracted. See Lesson 1-7.

8. a. b. b. 1.5k2m 4x 2c3 Example 2 The degree is 3. Add the exponents of the variables: 2 + 1 = 3. The degree is 1. Add the exponents of the variables: 1 = 1. 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 polynomial is the degree of the term with the greatest degree.

10. Directions: Find the degree of each polynomial.

11. B. :degree 4 :degree 3 –5: degree 0 Example 3 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.

12. Example 4 a. 5x – 6 –6: degree 0 5x: degree 1 Find the degree of each term. The degree of the polynomial is the greatest degree, 1. b. x3y2 + x2y3 – x4 + 2 Find the degree of each term. x3y2: degree 5 x2y3: degree 5 –x4: degree 4 2: degree 0 The degree of the polynomial is the greatest degree, 5.

13. 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.

14. Write the polynomial in standard form. Then give the leading coefficient. Directions: Step 1: Find the degree of EACH term Step 2: Write in standard form by arranging terms in descending order Step 3: Determine the LEADING coefficient.

15. 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 5 6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in descending order:

16. 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 6 y2 + y6 − 3y Find the degree of each term. Then arrange them in descending order:

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

18. 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. Example 7 16 – 4x2 + x5 + 9x3 Find the degree of each term. Then arrange them in descending order:

19. 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. Example 8 18y5 – 3y8 + 14y Find the degree of each term. Then arrange them in descending order:

20. 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.

21. Classify each polynomial according to its degree and number of terms. Directions: Step 1: Determine degree of the polynomial or monomial (term with the greatest degree) – first name Step 2: Count how many total terms there are – last name

22. 4y6 – 5y3 + 2y – 9 is a 6th-degree polynomial. Example 9 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

23. 6 is a constant monomial. Example 10 a. x3 + x2 – x + 2 x3 + x2 – x + 2 is acubic polymial. 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

24. Lesson Summary: Part I Find the degree of each polynomial. 1. 7a3b2 – 2a4 + 4b –15 2. 25x2 – 3x4 Write each polynomial in standard form. Then give the leading coefficient. 3. 24g3 + 10 + 7g5 – g2 4. 14 – x4 + 3x2 5 4 7g5 + 24g3 – g2 + 10; 7 –x4 + 3x2 + 14; –1

25. Lesson Summary: Part II Classify each polynomial according to its degree and number of terms. quadratic trinomial 5. 18x2 – 12x + 5 6. 2x4 – 1 quartic binomial