1. Chapter 7 – Exponents and Polynomials Algebra I

2. Table of Contents 7-7 - Adding and Subtracting Polynomials 7-7 7-8 - Multiplying Polynomials 7-8 7-9 - Special Products of Binomials 7-9

3. 7.7 - Adding and Subtracting Polynomials Algebra I

4. 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. (5x 2 + 4x + 1) + (2x 2 + 5x + 2) = (5x 2 + 2x 2 ) + (4x + 5x) + (1 + 2) = 7x 2 + 9x + 3 5x 2 + 4x + 1 + 2x 2 + 5x + 2 7x2 + 9x + 37x2 + 9x + 3 7.7 Algebra 1 (bell work)

5. Add or Subtract. By combining like terms A. 12p 3 + 11p 2 + 8p 3 12p 3 + 11p 2 + 8p 3 12p 3 + 8p 3 + 11p 2 20p 3 + 11p 2 B. 5x 2 – 6 – 3x + 8 5x 2 – 6 – 3x + 8 5x 2 – 3x + 8 – 6 5x 2 – 3x + 2 7.7 Example 1Adding and Subtracting Monomials

6. Add or Subtract. By combining like terms (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 7.7 Example 2Adding Polynomials

7. Add or Subtract. By combining like terms 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 B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) (10xy – 3xy) + x + y 7xy + x + y 7.7

8. Math Joke Teacher: What is b + b? Shakespeare: Is it 2b or not 2b? 7.7

9. Add or Subtract. By combining like terms 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 7.7 b. 4z 4 – 8 + 16z 4 + 2 4z 4 – 8 + 16z 4 + 2 4z 4 + 16z 4 – 8 + 2 20z 4 – 6 Example 3Subtracting Polynomials

10. c. 2x 8 + 7y 8 – x 8 – y 8 Add or Subtract. By combining like terms 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 7.7

11. 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 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. (3x 2 + 7x – 5) (5x 2 – 4x + 11) 8x 2 + 3x + 6 Plot A. Plot B. Combine like terms. + 7.7

12. 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 profit. –0.02x 2 + 21x – 1700 Southern plant profit. Combine like terms. + –0.05x 2 + 46x – 3200 7.7

13. HW pg. 504 7.7- – 1 -15 (Odd), 16- 30 (Even), 34, 43, 44, 52, 64, 66- 72 – Ch: 53

14. 7.8 - Multiplying Polynomials Algebra I

15. Multiply A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) 18y 8 B. (3mn 2 ) (9m 2 n) (3mn 2 )(9m 2 n) 27m 3 n 3 (6 3)(y 3 y 5 )  (3 9)(m m 2 )(n 2  n)  7.8 Example 1Multiplying Monomials

16. Multiply     4522 1 12 3 xzyzx y    3  gggg    32245 1 z 3 zx y    755 4xyz

17. Multiply 4(3x 2 + 4x – 8) (4)3x 2 +(4)4x – (4)8 12x 2 + 16x – 32 7.8 Example 2Multiplying a Polynomial by a Monomial 3x 3 y 2 + 4x 4 y 3

18. Multiply 3ab(5a 2 + b) (3ab)(5a 2 ) + (3ab)(b) (3  5)(a  a 2 )(b) + (3)(a)(b  b) 15a 3 b + 3ab 2 7.8

19. Math Joke Student A: What is u times r times r? Student B: ur 2 Student A: No, I’m not!

20. To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3. Distribute x and 3 again. Multiply. Combine like terms. = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 7.8

22. (s + 4)(s – 2) s(s – 2) + 4(s – 2) s(s) + s(–2) + 4(s) + 4(–2) s 2 – 2s + 4s – 8 s 2 + 2s – 8 7.8 Example 3Multiplying Binomials (x – 4) 2 (x – 4)(x – 4) (x x) + (x (–4)) + (–4  x) + (–4  (–4))  x 2 – 4x – 4x + 16 x 2 – 8x + 16

23. Multiply (8m 2 – n)(m 2 – 3n) 8m 2 (m 2 ) + 8m 2 (–3n) – n(m 2 ) – n(–3n) 8m 4 – 24m 2 n – m 2 n + 3n 2 8m 4 – 25m 2 n + 3n 2 7.8 (2a – b 2 )(a + 4b 2 ) 2a(a) + 2a(4b 2 ) – b 2 (a) + (–b 2 )(4b 2 ) 2a 2 + 8ab 2 – ab 2 – 4b 4 2a 2 + 7ab 2 – 4b 4

24. Multiply (x – 5)(x 2 + 4x – 6) x(x 2 + 4x – 6) – 5(x 2 + 4x – 6) x(x 2 ) + x(4x) + x(–6) – 5(x 2 ) – 5(4x) – 5(–6) x 3 + 4x 2 – 5x 2 – 6x – 20x + 30 x 3 – x 2 – 26x + 30 7.8 Day 2Example 4Multiplying Polynomials

25. Multiply (2x – 5)(–4x 2 – 10x + 3) 7.8

26. Multiply (x + 3) 3 [(x + 3)(x + 3)](x + 3) [x(x+3) + 3(x+3)](x + 3) (x 2 + 3x + 3x + 9)(x + 3) (x 2 + 6x + 9)(x + 3) 7.8 x 3 + 9x 2 + 27x + 27 Final Answer x 3 + 6x 2 + 9x + 3x 2 + 18x + 27 x(x 2 ) + x(6x) + x(9) + 3(x 2 ) + 3(6x) + 3(9) x(x 2 + 6x + 9) + 3(x 2 + 6x + 9) (x + 3)(x 2 + 6x + 9)

27. The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. a. Write a polynomial that represents the area of the base of the prism. A = l  w A = l w  A = (h + 4)(h – 3) A = h 2 + 4h – 3h – 12 A = h 2 + h – 12 The area is represented by h 2 + h – 12. 7.8 Example 5Application

28. The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. b. Find the area of the base when the height is 5 ft. A = h 2 + h – 12 A = 5 2 + 5 – 12 A = 25 + 5 – 12 A = 18 The area is 18 square feet. 7.8

29. HW pg. 517 7.8- – Day 1: 1-17 (Odd), 102-104 – Day 2: 19-25 (Odd), 56, 62, 66-68 – Ch: 64

30. 7.9 - Special Products of Binomials Algebra I

31. We will have 3 different types of problems in this section You have done each type of these 3 problems, but by using certain patterns These can lead to quicker ways of solving or simplifying 7.9Algebra 1 (bell work) Pg. 524

32. A. (x +3) 2 (a + b) 2 = a 2 + 2ab + b 2 (x + 3) 2 = x 2 + 2(x)(3) + 3 2 = x 2 + 6x + 9 B. (4s + 3t) 2 (a + b) 2 = a 2 + 2ab + b 2 (4s + 3t) 2 = (4s) 2 + 2(4s)(3t) + (3t) 2 = 16s 2 + 24st + 9t 2 7.9Example 1Finding Products in the Form (a + b) 2

33. Multiply C. (5a + b) 2 (a + b) 2 = a 2 + 2ab + b 2 (5a + b) 2 = (5a) 2 + 2(5a)(b) + b 2 = 25a 2 + 10ab + b 2 7.9 Optional

34. You can use the FOIL method to find products in the form of (a – b) 2. (a – b) 2 = (a – b)(a – b) = a 2 – ab – ab + b 2 F L I O = a 2 – 2ab + b 2 A trinomial of the form a 2 – ab + b 2 is also a perfect-square trinomial because it is the result of squaring the binomial (a – b). 7.9Just Read

35. Multiply A. (x – 6) 2 (a – b) 2 = a 2 – 2ab + b 2 (x – 6) 2 = x 2 – 2x(6) + (6) 2 = x 2 – 12x + 36 B. (4m – 10) 2 (a – b) 2 = a 2 – 2ab + b 2 (4m – 10) 2 = (4m) 2 – 2(4m)(10) + (10) 2 = 16m 2 – 80m + 100 7.9Example 2Finding Products in the Form (a – b) 2

36. Multiply C. (2x – 5y) 2 (a – b) 2 = a 2 – 2ab + b 2 (2x – 5y) 2 = (2x) 2 – 2(2x)(5y) + (5y) 2 = 2x 2 – 20xy +25y 2 D. (7 – r 3 ) 2 (a – b) 2 = a 2 – 2ab + b 2 (7 – r 3 ) 2 = 7 2 – 2(7)(r 3 ) + (r 3 ) 2 = 49 – 14r 3 + r 6 7.9Optional

37. Math Joke Parent: What happened in math class today? Student: When the teacher said to look for the perfect square, everyone looked at me

38. Multiply A. (x + 4)(x – 4) (a + b)(a – b) = a 2 – b 2 (x + 4)(x – 4) = x 2 – 4 2 = x 2 – 16 B. (p 2 + 8q)(p 2 – 8q) (a + b)(a – b) = a 2 – b 2 (p 2 + 8q)(p 2 – 8q) = p 4 – (8q) 2 = p 4 – 64q 2 7.9Example 3Finding Products in the Form ( a + b)(a – b)

39. Multiply C. (10 + b)(10 – b) (a + b)(a – b) = a 2 – b 2 (10 + b)(10 – b) = 10 2 – b 2 = 100 – b 2 7.9Optional

40. Multiply a. (x + 8)(x – 8) (a + b)(a – b) = a 2 – b 2 (x + 8)(x – 8) = x 2 – 8 2 = x 2 – 64 b. (3 + 2y 2 )(3 – 2y 2 ) (a + b)(a – b) = a 2 – b 2 (3 + 2y 2 )(3 – 2y 2 ) = 3 2 – (2y 2 ) 2 = 9 – 4y 4 7.9Optional

41. HW pg. 525 7.9- – 3-19 (Odd), 20, 76-82 – Ch: 39, 40, 64-66