7-5

1. 7-5 Polynomials Lesson Presentation Lesson Quiz Holt Algebra 1

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

3. Objective Add and subtract polynomials.

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 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.

6. A. 4p4q3 Example 1: Finding the Degree of a Monomial Find the degree of each monomial. 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 Check It Out! Example 1 Find the degree of each monomial. 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. 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.

11. Check It Out! Example 2 Find the degree of each polynomial. 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.

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 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.

13. 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:

14. 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:

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

16. 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:

17. 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:

18. 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.

19. 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

20. 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 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

21. Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

22. Example 1: Adding and Subtracting Monomials Add or Subtract.. A. 12p3 + 11p2 + 8p3 Identify like terms. 12p3 + 11p2 + 8p3 Rearrange terms so that like terms are together. 12p3 + 8p3 + 11p2 20p3 + 11p2 Combine like terms. B. 5x2 – 6 – 3x + 8 Identify like terms. 5x2– 6 – 3x+ 8 Rearrange terms so that like terms are together. 5x2 – 3x+ 8 – 6 5x2 – 3x + 2 Combine like terms.

23. Example 1: Adding and Subtracting Monomials Add or Subtract.. C. t2 + 2s2 – 4t2 – s2 Identify like terms. t2+ 2s2– 4t2 – s2 Rearrange terms so that like terms are together. t2– 4t2+ 2s2 – s2 –3t2+ s2 Combine like terms. D. 10m2n + 4m2n – 8m2n 10m2n + 4m2n – 8m2n Identify like terms. 6m2n Combine like terms.

24. Remember! Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7.

25. Check It Out! Example 1 Add or subtract. a. 2s2 + 3s2 + s 2s2 + 3s2 + s Identify like terms. 5s2 + s Combine like terms. b. 4z4 – 8 + 16z4 + 2 Identify like terms. 4z4– 8+ 16z4+ 2 Rearrange terms so that like terms are together. 4z4+ 16z4– 8+ 2 20z4 – 6 Combine like terms.

26. Check It Out! Example 1 Add or subtract. c. 2x8 + 7y8 – x8 – y8 Identify like terms. 2x8+ 7y8– x8– y8 Rearrange terms so that like terms are together. 2x8– x8+ 7y8– y8 x8 + 6y8 Combine like terms. d. 9b3c2 + 5b3c2 – 13b3c2 Identify like terms. 9b3c2 + 5b3c2 – 13b3c2 b3c2 Combine like terms.

27. 5x2+ 4x+1 + 2x2+ 5x+ 2 7x2+9x+3 Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x2 + 4x + 1) + (2x2 + 5x+ 2) = (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2) = 7x2+ 9x+ 3

28. Example 2: Adding Polynomials Add. A. (4m2 + 5) + (m2 – m + 6) (4m2+ 5) + (m2– m + 6) Identify like terms. Group like terms together. (4m2+m2) + (–m)+(5 + 6) 5m2 – m + 11 Combine like terms. B. (10xy + x) + (–3xy + y) Identify like terms. (10xy + x) + (–3xy + y) Group like terms together. (10xy– 3xy) + x +y 7xy+ x +y Combine like terms.

29. 6x2– 4y + –5x2+ y Example 2C: Adding Polynomials Add. (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) (6x2– 4y) + (3x2+ 3y –8x2– 2y) Identify like terms. Group like terms together within each polynomial. (6x2 + 3x2 – 8x2) + (3y – 4y – 2y) Use the vertical method. Combine like terms. x2– 3y Simplify.

30. Example 2D: Adding Polynomials Add. Identify like terms. Group like terms together. Combine like terms.

31. Check It Out! Example 2 Add (5a3 + 3a2 – 6a + 12a2) + (7a3–10a). (5a3+ 3a2 – 6a+ 12a2) + (7a3–10a) Identify like terms. Group like terms together. (5a3+ 7a3)+ (3a2+ 12a2) + (–10a – 6a) 12a3 + 15a2 –16a Combine like terms.

32. To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: –(2x3 – 3x + 7)= –2x3 + 3x– 7

33. Example 3A: Subtracting Polynomials Subtract. (x3 + 4y) – (2x3) Rewrite subtraction as addition of the opposite. (x3 + 4y) + (–2x3) (x3 + 4y) + (–2x3) Identify like terms. (x3– 2x3) + 4y Group like terms together. –x3 + 4y Combine like terms.

34. Example 3B: Subtracting Polynomials Subtract. (7m4 – 2m2) – (5m4 – 5m2 + 8) (7m4 – 2m2) + (–5m4+5m2– 8) Rewrite subtraction as addition of the opposite. (7m4– 2m2) + (–5m4+ 5m2 – 8) Identify like terms. Group like terms together. (7m4– 5m4) + (–2m2+ 5m2) – 8 2m4 + 3m2 – 8 Combine like terms.

35. –10x2 – 3x + 7 –x2 + 0x+ 9 Example 3C: Subtracting Polynomials Subtract. (–10x2 – 3x + 7) – (x2 – 9) (–10x2 – 3x + 7) + (–x2+9) Rewrite subtraction as addition of the opposite. (–10x2 – 3x + 7) + (–x2+ 9) Identify like terms. Use the vertical method. Write 0x as a placeholder. –11x2 – 3x + 16 Combine like terms.

36. 9q2 – 3q+ 0 +− q2– 0q + 5 Example 3D: Subtracting Polynomials Subtract. (9q2 – 3q) – (q2 – 5) Rewrite subtraction as addition of the opposite. (9q2 – 3q) + (–q2+ 5) (9q2 – 3q) + (–q2 + 5) Identify like terms. Use the vertical method. Write 0 and 0q as placeholders. 8q2 – 3q + 5 Combine like terms.

37. –x2+ 0x + 1 + –x2 – x – 1 Check It Out! Example 3 Subtract. (2x2 – 3x2 + 1) – (x2+ x + 1) Rewrite subtraction as addition of the opposite. (2x2 – 3x2 + 1) + (–x2– x – 1) (2x2– 3x2+ 1) + (–x2 – x – 1) Identify like terms. Use the vertical method. Write 0x as a placeholder. –2x2– x Combine like terms.