1. 7-7 Adding and Subtracting Polynomials Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 7-7 Adding and Subtracting Polynomials Warm Up Simplify each expression by combining like terms. 1. 4x + 2x 2. 3y + 7y 3. 8p – 5p 4. 5n + 6n 2 Simplify each expression. 5. 3(x + 4) 6. –2(t + 3) 7. –1(x 2 – 4x – 6) 6x6x 10y 3p3p not like terms 3x + 12 –2t – 6 –x 2 + 4x + 6

3. 7-7 Adding and Subtracting Polynomials 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques. California Standards

4. 7-7 Adding and Subtracting Polynomials Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

5. 7-7 Adding and Subtracting Polynomials Add or subtract. Additional Example 1: Adding and Subtracting Monomials A. 12p 3 + 11p 2 + 8p 3 12p 3 + 11p 2 + 8p 3 12p 3 + 8p 3 + 11p 2 20p 3 + 11p 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. B. 5x 2 – 6 – 3x + 8 5x 2 – 6 – 3x + 8 5x 2 – 3x + 8 – 6 5x 2 – 3x + 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

6. 7-7 Adding and Subtracting Polynomials Add or subtract. Additional Example 1: Adding and Subtracting Monomials C. t 2 + 2s 2 – 4t 2 – s 2 t 2 – 4t 2 + 2s 2 – s 2 t 2 + 2s 2 – 4t 2 – s 2 –3t 2 + s 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. D. 10m 2 n + 4m 2 n – 8m 2 n 10m 2 n + 4m 2 n – 8m 2 n 6m2n6m2n Identify like terms. Combine like terms.

7. 7-7 Adding and Subtracting Polynomials 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. Remember!

8. 7-7 Adding and Subtracting Polynomials Check It Out! Example 1 a. 2s 2 + 3s 2 + s Add or subtract. 2s 2 + 3s 2 + s 5s 2 + s b. 4z 4 – 8 + 16z 4 + 2 4z 4 – 8 + 16z 4 + 2 4z 4 + 16z 4 – 8 + 2 20z 4 – 6 Identify like terms. Combine like terms. Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

9. 7-7 Adding and Subtracting Polynomials Check It Out! Example 1 c. 2x 8 + 7y 8 – x 8 – y 8 Add or subtract. 2x 8 + 7y 8 – x 8 – y 8 2x 8 – x 8 + 7y 8 – y 8 x 8 + 6y 8 d. 9b 3 c 2 + 5b 3 c 2 – 13b 3 c 2 9b 3 c 2 + 5b 3 c 2 – 13b 3 c 2 b3c2 b3c2 Identify like terms. Combine like terms. Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

10. 7-7 Adding and Subtracting Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: 5x 2 + 4x + 1 + 2x 2 + 5x + 2 7x2 + 9x + 37x2 + 9x + 3

11. 7-7 Adding and Subtracting Polynomials In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x 2 + 4x + 1) + (2x 2 + 5x + 2) = (5x 2 + 2x 2 ) + (4x + 5x) + (1 + 2) = 7x 2 + 9x + 3

12. 7-7 Adding and Subtracting Polynomials Add. Additional Example 2: Adding Polynomials A. (4m 2 + 5) + (m 2 – m + 6) (4m 2 + 5) + (m 2 – m + 6) (4m 2 + m 2 ) + (–m) + (5 + 6) 5m 2 – m + 11 Identify like terms. Group like terms together. Combine like terms. B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) (10xy – 3xy) + x + y 7xy + x + y Identify like terms. Group like terms together. Combine like terms.

13. 7-7 Adding and Subtracting Polynomials Add. Additional Example 2: Adding Polynomials C. (6x 2 – 4y) + (3x 2 + 3y – 8x 2 – 2y) Identify like terms. Group like terms together within each polynomial. Combine like terms. (6x 2 – 4y) + (3x 2 + 3y – 8x 2 – 2y) (6x 2 – 4y) + (3x 2 – 8x 2 + 3y – 2y) Use the vertical method. 6x 2 – 4y + –5x 2 + y x 2 – 3y Combine like terms in the second polynomial. (6x 2 – 4y) + (–5x 2 + y)

14. 7-7 Adding and Subtracting Polynomials Add. Additional Example 2: Adding Polynomials Identify like terms. Group like terms together. Combine like terms. D.

15. 7-7 Adding and Subtracting Polynomials When you use the Associative and Commutative Properties to rearrange the terms, the sign in front of each term must stay with that term. Writing Math

16. 7-7 Adding and Subtracting Polynomials Check It Out! Example 2 Add (5a 3 + 3a 2 – 6a + 12a 2 ) + (7a 3 – 10a). (5a 3 + 3a 2 – 6a + 12a 2 ) + (7a 3 – 10a) (5a 3 + 7a 3 ) + (3a 2 + 12a 2 ) + (–10a – 6a) 12a 3 + 15a 2 – 16a Identify like terms. Group like terms together. Combine like terms.

17. 7-7 Adding and Subtracting Polynomials 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: –(2x 3 – 3x + 7) = –2x 3 + 3x – 7

18. 7-7 Adding and Subtracting Polynomials Subtract. Additional Example 3A: Subtracting Polynomials (x 3 + 4y) – (2x 3 ) (x 3 + 4y) + (–2x 3 ) (x 3 – 2x 3 ) + 4y –x 3 + 4y Rewrite subtraction as addition of the opposite. Identify like terms. Group like terms together. Combine like terms.

19. 7-7 Adding and Subtracting Polynomials Subtract. Additional Example 3B: Subtracting Polynomials (7m 4 – 2m 2 ) – (5m 4 – 5m 2 + 8) (7m 4 – 2m 2 ) + (–5m 4 + 5m 2 – 8) (7m 4 – 5m 4 ) + (–2m 2 + 5m 2 ) – 8 (7m 4 – 2m 2 ) + (–5m 4 + 5m 2 – 8) 2m 4 + 3m 2 – 8 Rewrite subtraction as addition of the opposite. Identify like terms. Group like terms together. Combine like terms.

20. 7-7 Adding and Subtracting Polynomials Subtract. Additional Example 3C: Subtracting Polynomials (–10x 2 – 3x + 7) – (x 2 – 9) (–10x 2 – 3x + 7) + (–x 2 + 9) –10x 2 – 3x + 7 –x 2 + 0x + 9 –11x 2 – 3x + 16 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0x as a placeholder. Combine like terms.

21. 7-7 Adding and Subtracting Polynomials Subtract. Additional Example 3D: Subtracting Polynomials (9q 2 – 3q) – (q 2 – 5) (9q 2 – 3q) + (–q 2 + 5) 9q 2 – 3q + 0 + − q 2 – 0q + 5 8q 2 – 3q + 5 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0 and 0q as placeholders. Combine like terms.

22. 7-7 Adding and Subtracting Polynomials Check It Out! Example 3 Subtract. (2x 2 – 3x 2 + 1) – (x 2 + x + 1) (2x 2 – 3x 2 + 1) + (–x 2 – x – 1) –x 2 + 0x + 1 + –x 2 – x – 1 –2x 2 – x Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0x as a placeholder. Combine like terms.

23. 7-7 Adding and Subtracting Polynomials A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x 2 + 7x – 5, and the area of plot B can be represented by 5x 2 – 4x + 11. Write a polynomial that represents the total area of both plots of land. Additional Example 4: Application (3x 2 + 7x – 5) (5x 2 – 4x + 11) 8x 2 + 3x + 6 Plot A. Plot B. Combine like terms. +

24. 7-7 Adding and Subtracting Polynomials Check It Out! Example 4 The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant. Use the information above to write a polynomial that represents the total profits from both plants. –0.03x 2 + 25x – 1500 Eastern plant profits –0.02x 2 + 21x – 1700 Southern plant profits Combine like terms. + –0.05x 2 + 46x – 3200

25. 7-7 Adding and Subtracting Polynomials Lesson Quiz: Part I Add or subtract. 1. 7m 2 + 3m + 4m 2 2. (r 2 + s 2 ) – (5r 2 + 4s 2 ) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d 2 – 8) + (6d 2 – 2d + 1) –4r 2 – 3s 2 11m 2 + 3m 18pq – 2p 20d 2 – 2d – 7 5. (2.5ab + 14b) – (–1.5ab + 4b)4ab + 10b

26. 7-7 Adding and Subtracting Polynomials Lesson Quiz: Part II 6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x 2 + 12x + 9, and the area of the second wall is modeled by 36x 2 – 12x + 1. Write a polynomial that represents the total area of the two walls. 40x 2 + 10