Adding and Subtracting Polynomials

1. Adding and Subtracting Polynomials Lesson 8-1

2. Vocabulary • Monomial: a real number, or variable with a whole number as an exponent: • Binomial: two monomials being added or subtracted. • Trinomial: three monomials being added or subtracted. Label each as a monomial, binomial or trinomial. 5x2 6x2 + 4y7 + -2w3mn2 – 7y Degree of a monomial: the SUM of the exponents of its variables.

3. Problem 1 What is the degree of each monomial: • 5x • 6x3y2 • 4

4. Problem 2 What is the sum or difference? • 3x2 + 5x2 – x2 • 4x3y – x3y + 6x3y

5. Vocabulary Polynomial: a monomial or sum or monomial Degree of the polynomial: the variable with the largest exponent is also the degree 3x4 + 5x2 – 7x + 1 The degree is 4. Standard form is that you place the monomials in descending order from left to right.

7. Name that Polynomial! 3x + 4x2 Quadratic binomial 4x – 1 + 5x3 + 7x Cubic trinomial 2x – 3 – 8x2 Quadratic trinomial

8. Example 3 and 4 • It is exactly like combining like terms. • **Remember: 2x2 = 2x3 and they can not be combined.**

9. Problem 5 (x3 – 3x + 5x) - (7x3 + 5x2 – 12)

10. One More… (2g4 – 3g + 9) + (-g3 + 12g)

11. Multiplying and Factoring Lesson 8-2

12. 2x(3x + 1) = 6x2 + 2x x x x 1 x x x2 x x2 x2 x2 x2 x x2

13. What is –x3(9x4 – 2x3 + 7)? -x3(9x4) + (-x3)(-2x3) + (-x3)(7) -9x7 + 2x6 + -7x3

14. Got it? What is the simpler form of 5n(3n3 – n2 + 8)? 5n(3n3) – 5n(n2) + 5n(8) 15n4 – 5n3 + 40n

15. Factoring • Factoring is the opposite of multiplying. • It “undoes” what we did in the last slide.

16. What is the GCF of 5x3 + 25x2 + 45x? 5x3 = 5  x  x  x 25x2 = 5  5  x  x 45x = 3  3  5  x What do they have in common? 5x

17. Got it? What is the GCF of 3x4 – 9x2 – 12x? 3x4 = 3  x  x  x  x -9x2= -1  3  3  x  x -12x = -1  3  4  x What do they have in common? 3x

18. Factor 4x5 – 24x3 + 8x Common(leftover + leftover + leftover) 4x5 = 2 ∙ 2 ∙ x ∙ x ∙ x ∙ x ∙ x -24x3 = -1 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ x ∙ x ∙ x 8x = 2 ∙ 2 ∙ 2 ∙ x Common: 2 ∙ 2 ∙ x Left over: x4 – 6x2 + 2 Final Answer: 4x(x4 – 6x2 + 2)

19. Got it? Factor 9x6 + 15x4 + 12x2 9x6 = 3  3  x  x  x x x  x 15x4 = 3  5  x  x  x  x 12x2 = 2  2  3  x  x Common: 3x2 Leftover: 3x4 + 5x2 + 4 Final Answer: 3x2(3x4 + 5x2 + 4)

20. Example 4

21. Got it?

22. Multiplying Binomials Lesson 8-3

23. Multiply: (2x + 1)(x + 2) x x 1 x 2 2x2 + 5x + 2

24. Multiply: (2x + 4)(3x - 7) • Use the distributive property: 2x(3x - 7) + 4(3x - 7) 6x2 - 14x + 12x - 28 6x2 - 2x - 28

25. Got it? Multiply: (x – 6)(4x + 3) • Use the distributive property: x(4x + 3) + -6(4x + 3) 4x2 + 3x + -24x + -18 4x2– 21x + -18

26. Multiply: (x - 3)(4x - 5) • Make a table: 4x2 – 12x – 5x + 15 4x2 – 17x + 15

27. Multiply: (3x + 1)(x + 4) • Make a table: 3x2 –x – 12x + 4 3x2 – 13x + 4

28. Multiply: (5x – 3)(2x + 1) • Use FOIL (First, Outside, Inside, Last) (5x – 3)(2x + 1) (5x)(2x) + (5x)(1) + (-3)(2x) + (-3)(1) 10x2+5x– 6x – 3 10x2– x – 3

29. Got it? Multiply: (3x - 4)(x + 2) • Use FOIL (First, Outside, Inside, Last) (3x – 4)(x + 2) (3x)(x) + (3x)(2) + (-4)(x) + (-4)(2) 3x2+6x– 4x – 8 3x2 + 2x – 8

30. Application A cylinder has the dimensions shown in the diagram. Which polynomial in standard form best describes the total surface area of a cylinder with a radius of (x + 1) and a height of (x + 4)? The formula for a cylinder is 2πr2 + 2πrh, where r is the radius and h is the height.

31. Application 2πr2 + 2πrh 2π(x + 1)2+ 2π(x + 1)(x + 4) 2π(x + 1)(x + 1)+ 2π (x + 1)(x + 4) 2π(x2 + 2x + 1) + 2π(x2 + 5x + 4) 2π (x2 + 2x + 1+x2 + 5x + 4) 2π(2x2+ 7x + 5) 4πx2 + 14πx + 10π

32. Let’s do number 29 or 30 together….

33. Multiplying Special Cases Lesson 8-4

34. The Square of a Binomial (a + b)2 = a2 + 2ab + b2 (x + 6)2 = x2 + 2(x)(6) + 62 = x2 + 12x + 36 (a – b)2= a2 – 2ab + b2 (x - 5)2= x2– 2(x)(5) + 52 = x2– 10x + 25

35. Multiply: (x + 3)2 x 3 x 3 x2 + 3x + 3x + 9 x2 + 6x + 9

36. Problem 1a: Multiply: (x + 8)2 (x + 8) = x2 + 2(x)(8) + 82 = x2 + 16x + 64 Using FOIL: (x + 8)2 = (x + 8)(x + 8) x2 + 8x + 8x + 64 x2 + 16x + 64

37. Got it? Multiply: (x + 12)2 x2+ 24x+ 144

38. Problem 1b: Multiply: (x - 7)2 (x - 7) = x2 - 2(x)(7) + 72 = x2 - 14x + 49 Using FOIL: (x - 7)2 = (x - 7)(x - 7) x2 - 7x - 7x + 49 x2- 14x + 49

39. Got it? Multiply: (2x - 9)2 (2x)2 – 2(2x)(9) + 92 4x2 - 36x+ 81

40. Problem 2: Applying Squares of Binomials A square patio is surrounded by the brick walkway shown. What is the area of the walkway?

41. Problem 2: Applying Squares of Binomials (Continued) Total Area: (x + 6)2 = x2 + 2(x)(6) + 62 Total Area = x2 + 12x + 36 Area of patio: x ∙ x = x2 Area of walkway = Total Area – Patio = x2 + 12x + 36 – x2 = 12x + 36

42. Problem 3: Using Mental Math What is 392? Use mental math. 392 = (40 – 1)2 (40 – 1)2 = 402 – 2(40(1) + 12 = 1600 – 80 – 1 = 1521

43. Got it? Use mental math to compute 852. 7225

44. What if…. (a + b)(a – b)? (a + b)(a – b) = a2 – b2 (x + 2)(x – 2) = x2 – 4 Using FOIL: (x + 2)(x – 2) x2 - 2x + 2x + -4 x2 – 2

45. Problem 4: (x + 5)(x – 5) = x2 – 52 = x2 – 25 (x3 + 8)(x3 – 8) (x3)2 – 82 x6 - 64

46. Got it? (x + 9)(x – 9) x2 - 81 (3c – 4)(3c + 4) 9c2 - 16

47. Problem 5 What is 64 x 56 using mental math? (60 + 4)(60 – 4) 602 – 42 3600 – 16 3584

48. Factoring x2 + bx + c Lesson 8-5

49. Key Concept: (x + 3)(x + 7) (x + 3)(x + 7) (x)(x) + 3x + 7x + (3)(7) x2+ (3 + 7)x + 21 10 is 3 + 7 and 21 is 3 7

50. Problem 1: Factor x2 + 8x + 15 • Ask: what are the addition factors of 8? (whole numbers only) • What are the multiplication factors of 15? • When are they the same? • 8 = (1 + 7) (2 + 6) (3 + 5) (4 + 4) • 15 = (1 x 15) (3 x 5) • 3 and 5 appear in both. (x + 3)(x + 5)