1. Algebra 1B Chapter 10 Polynomials and Factoring Adding and Subtracting

2. 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. To add or subtract polynomials, combine like terms.

4. 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.

5. 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.

6. In Your Notes! 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.

7. In Your Notes! 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.

8. 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

9. 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

10. 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.

11. 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)

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

13. 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

14. In Your Notes! 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.

15. 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

16. 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.

17. 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.

18. 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.

19. 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.

20. In Your Notes! 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.

21. 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. +

22. In Your Notes! 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