Adding and Subtracting Polynomials

1. Algebra 1: Chapter 7 Exponents and Polynomials 7.7 Adding and Subtracting Polynomials

2. California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.

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

4. Additional 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.

5. Additional 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.

6. 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.

7. 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.

8. 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.

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

10. In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x2 + 4x + 1) + (2x2 + 5x+ 2) = (5x2 + 2x2) + (4x + 5x) + (1 + 2) = 7x2+ 9x+ 3

11. Additional 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.

12. 6x2– 4y + –5x2+ y Additional Example 2: Adding Polynomials Add. C. (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) (6x2– 4y) + (3x2+ 3y –8x2– 2y) Identify like terms. Group like terms together within each polynomial. (6x2– 4y) +(3x2– 8x2 + 3y – 2y) (6x2– 4y) +(–5x2 + y) Combine like terms in the second polynomial. Use the vertical method. x2– 3y Combine like terms.

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

14. Writing Math When you use the Associative and Commutative Properties to rearrange the terms, the sign in front of each term must stay with that term.

15. 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.

16. 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

17. Additional 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.

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

19. –10x2 – 3x + 7 –x2 + 0x+ 9 Additional 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.

20. 9q2 – 3q+ 0 +− q2– 0q + 5 Additional 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.

21. –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.

22. 8x2 + 3x + 6 Additional Example 4: Application 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 3x2 + 7x – 5, and the area of plot B can be represented by 5x2 – 4x + 11. Write a polynomial that represents the total area of both plots of land. (3x2 + 7x – 5) Plot A. (5x2– 4x + 11) Plot B. + Combine like terms.

23. –0.05x2 + 46x – 3200 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. –0.03x2 + 25x – 1500 Eastern plant profits + –0.02x2 + 21x – 1700 Southern plant profits Combine like terms.

24. 7.7 Lesson Quiz Add or subtract. 1. 7m2 + 3m + 4m2 2. (r2 + s2) – (5r2 + 4s2) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d2 – 8) + (6d2 – 2d + 1) 11m2 + 3m –4r2 – 3s2 18pq – 2p 20d2 – 2d – 7 5. (2.5ab + 14b) – (–1.5ab + 4b) 4ab + 10b 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 4x2 + 12x + 9, and the area of the second wall is modeled by 36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls. 40x2 + 10