Polynomials and Geometry

1. Section 11-2
Polynomials and Geometry
Sunday, March 1, 2009

2. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
Sunday, March 1, 2009

3. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh
Sunday, March 1, 2009

4. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9)
Sunday, March 1, 2009

5. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9) = 2457 mm3
Sunday, March 1, 2009

6. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9) = 2457 mm3
SA = 2lw + 2lh + 2hw
Sunday, March 1, 2009

7. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9) = 2457 mm3
SA = 2lw + 2lh + 2hw
= 2(21)(13) + 2(21)(9) + 2(13)(9)
Sunday, March 1, 2009

8. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9) = 2457 mm3
SA = 2lw + 2lh + 2hw
= 2(21)(13) + 2(21)(9) + 2(13)(9)
= 546 + 378 + 234
Sunday, March 1, 2009

9. Warm-up
A box is 21 mm long, 13 mm wide, and 9
mm high. Find the volume and surface area.
V = lwh = 21(13)(9) = 2457 mm3
SA = 2lw + 2lh + 2hw
= 2(21)(13) + 2(21)(9) + 2(13)(9)
= 546 + 378 + 234
= 1158 mm2
Sunday, March 1, 2009

10. 3x
3x + 3
3x2 + 3x + 3
Anything else:
Sunday, March 1, 2009

11. 3x Monomial
3x + 3
3x2 + 3x + 3
Anything else:
Sunday, March 1, 2009

12. 3x Monomial (One term)
3x + 3
3x2 + 3x + 3
Anything else:
Sunday, March 1, 2009

13. 3x Monomial (One term)
3x + 3 Binomial
3x2 + 3x + 3
Anything else:
Sunday, March 1, 2009

14. 3x Monomial (One term)
3x + 3 Binomial (Two terms)
3x2 + 3x + 3
Anything else:
Sunday, March 1, 2009

15. 3x Monomial (One term)
3x + 3 Binomial (Two terms)
3x2 + 3x + 3 Trinomial
Anything else:
Sunday, March 1, 2009

16. 3x Monomial (One term)
3x + 3 Binomial (Two terms)
3x2 + 3x + 3 Trinomial (Three terms)
Anything else:
Sunday, March 1, 2009

17. 3x Monomial (One term)
3x + 3 Binomial (Two terms)
3x2 + 3x + 3 Trinomial (Three terms)
Anything else: Polynomial
Sunday, March 1, 2009

18. 3x Monomial (One term)
3x + 3 Binomial (Two terms)
3x2 + 3x + 3 Trinomial (Three terms)
Anything else: Polynomial (Many terms)
Sunday, March 1, 2009

19. x4 + 3x2y3 + y4
Sunday, March 1, 2009

20. x4 + 3x2y3 + y4
x4
Sunday, March 1, 2009

21. x4 + 3x2y3 + y4
x4 3x2y3
Sunday, March 1, 2009

22. x4 + 3x2y3 + y4
x4 3x2y3 y4
Sunday, March 1, 2009

23. x4 + 3x2y3 + y4
x4 3x2y3 y4
4 x’s
Sunday, March 1, 2009

24. x4 + 3x2y3 + y4
x4 3x2y3 y4
2 x’s
4 x’s
3 y’s
Sunday, March 1, 2009

25. x4 + 3x2y3 + y4
x4 3x2y3 y4
2 x’s
4 x’s 4 y’s
3 y’s
Sunday, March 1, 2009

26. x4 + 3x2y3 + y4
x4 3x2y3 y4
2 x’s
4 x’s 4 y’s
3 y’s
Sunday, March 1, 2009

27. x4 + 3x2y3 + y4
x4 3x2y3 y4
2 x’s
4 x’s 4 y’s
3 y’s
This has 5 variables
Sunday, March 1, 2009

28. x4 + 3x2y3 + y4
x4 3x2y3 y4
2 x’s
4 x’s 4 y’s
3 y’s
This has 5 variables
The degree of this polynomial is 5.
Sunday, March 1, 2009

29. Degree of a polynomial in several variables
Sunday, March 1, 2009

30. Degree of a polynomial in several variables
The largest sum of the exponents of the variables in
the terms.
Sunday, March 1, 2009

31. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
z+3
x y
Sunday, March 1, 2009

32. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
z+3
x y
Sunday, March 1, 2009

33. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b=
z+3
x y
Sunday, March 1, 2009

34. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
x y
Sunday, March 1, 2009

35. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
h=
x y
Sunday, March 1, 2009

36. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
h= z+3
x y
Sunday, March 1, 2009

37. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
h= z+3
x y
Sunday, March 1, 2009

38. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
h= z+3
x y
Sunday, March 1, 2009

39. Example 1
Find a polynomial for A (the area of a triangle) in
terms of x, y, and z.
b= x+y
z+3
h= z+3
x y
Sunday, March 1, 2009

40. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

41. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

42. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

43. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

44. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

45. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

46. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

47. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
= 6x3
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

48. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
= 6x3 - 13x2
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

49. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
= 6x3 - 13x2 + 13x
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

50. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
= 6x3 - 13x2 + 13x - 4
b. (y2 + 2y - 5)(4y2 - 6y -1)
Sunday, March 1, 2009

51. Example 2
Expand and simplify.
a. (2x - 1)(3x2 - 5x + 4)
= 6x3 - 10x2 + 8x - 3x2 + 5x - 4
= 6x3 - 13x2 + 13x - 4
b. (y2 + 2y - 5)(4y2 - 6y -1)
= 4y4 + 2y3 - 33y2 + 28y + 5
Sunday, March 1, 2009

52. Extended distribution:
Sunday, March 1, 2009

53. Extended distribution:
Multiply each term in the first polynomial by
each term in the second polynomial.
Sunday, March 1, 2009

54. Example 3
A piece of cardboard is used to make an open
box. It is 16.5 in by 23.5 in, with corners x in by
x in cut from each corner. Let V(x) be the volume
of the box. Find a polynomial to represent the
volume.
Sunday, March 1, 2009

55. Sunday, March 1, 2009

56. Sunday, March 1, 2009

57. Sunday, March 1, 2009

58. Sunday, March 1, 2009

59. 16.5 - 2x
Sunday, March 1, 2009

60. 23.5 - 2x
16.5 - 2x
Sunday, March 1, 2009

61. 23.5 - 2x
x
x
16.5 - 2x
Sunday, March 1, 2009

62. 23.5 - 2x
x
x
16.5 - 2x
V = lwh
Sunday, March 1, 2009

63. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)
Sunday, March 1, 2009

64. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)
Sunday, March 1, 2009

65. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
Sunday, March 1, 2009

66. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75
Sunday, March 1, 2009

67. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x
Sunday, March 1, 2009

68. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x - 47x
Sunday, March 1, 2009

69. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x - 47x + 4x2)
Sunday, March 1, 2009

70. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x - 47x + 4x2)(x)
Sunday, March 1, 2009

71. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x - 47x + 4x2)(x)
= 4x3 - 80x2 +387.75x
Sunday, March 1, 2009

72. 23.5 - 2x
x
x
16.5 - 2x
V = lwh = (16.5 - 2x)(23.5 - 2x)(x)
= (387.75 - 33x - 47x + 4x2)(x)
= 4x3 - 80x2 +387.75x in3
Sunday, March 1, 2009

73. Homework
Sunday, March 1, 2009

74. Homework
p. 683 #1-20
quot;The best doctor in the world is the
veterinarian. He can't ask his patients what is
the matter-he's got to just know.quot; – Will Rogers
Sunday, March 1, 2009