1. Unit 6: Polynomials 6. 1 Objectives The student will be able to:
1. multiply monomials.
simplify expressions with monomials.
Hernandez – Henry Ford High School

2. A monomial is a 1. number, 2. variable, or
3. a product of one or more numbers and variables.
Examples: 5 y
3x2y3

3. Why are the following not monomials? x + y
addition
division
2 - 3a
subtraction

4. Multiplying Monomials
When multiplying monomials, you ADD the exponents.
1) x2 • x4
x2+4 x6
2) 2a2y3 • 3a3y4
6a5y7

5. Simplify m3(m4)(m) m7 m8
m12
m13

6. Power of a Power
When you have an exponent with an exponent, you multiply those exponents.
1) (x2)3
x2• 3 x6
2) (y3)4
y12

7. Simplify (p2)4 p2 p4 p8
p16

8. Power of a Product
When you have a power outside of the parentheses, everything in the parentheses is raised to that power.
1) (2a)3
23a3
8a3
2) (3x)2
9x2

9. Simplify (4r)3
12r3
12r4
64r3
64r4

10. This is a combination of all of the other rules.
Power of a Monomial
This is a combination of all of the other rules.
1) (x3y2)4
x3• 4 y2• 4
x12 y8
2) (4x4y3)3
64x12y9

11. Simplify (3a2b3)4
12a8b12
81a6b7
81a16b81
81a8b12

12. 6.2 Objectives The student will be able to:
1. divide monomials.
simplify negative exponents.

13. When dividing monomials, subtract the exponents.1. 2.
= b5-2
= b3
= m7-1n5-2
= m6 n3

14. = xy
= 9a3b2

15. Simplify
48g2h2
48gh2
4g2h2
4gh2

16. = 1m0n = n Here’s a tricky one! They canceled out!
What happened to the m?
They canceled out!
There are no m’s left over!
This leads us to our next rule…

17. Zero Exponents Anything to the 0 power is equal to 1. a0 = 1
True or False? Anything divided by itself equals one.
True!
See for yourself!

18. Negative Exponents
A negative exponent means you move the base to the other side of the fraction and make the exponent positive.
Notice that the base with the negative exponent moved and became positive!

19. Simplify.
x-4 y0 You can not have negative or zero exponents in your answer.

20. Simplify p2
p12 .

21. Simplify.
You can’t leave the negative exponent!
There is another way of doing this without negative exponents.
If you don’t want to see it, skip the next slide!!!

22. Simplify (alternate version).
Look and see (visualize) where you have the larger exponent and leave the variable in that location. Subtract the smaller exponent from the larger one.
In this problem, r is larger in the numerator and s is larger in the denominator.
Notice that you did not have to work with negative exponents! This method is quicker!

23. Simplify.
Get rid of the negative exponent.

24. Simplify.
Get rid of the negative exponents.

25. Get rid of the negative exponents.
Simplify.
Get rid of the parentheses.
Get rid of the negative exponents.

26. Simplify .

27. 6.3 Objective The student will be able to:
express numbers in scientific and decimal notation.

28. How wide is our universe?
210,000,000,000,000,000,000,000 miles
(22 zeros)
This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

29. A number is expressed in scientific notation when it is in the form
a x 10n
where a is between 1 and 10
and n is an integer

30. Write the width of the universe in scientific notation.
210,000,000,000,000,000,000,000 miles
Where is the decimal point now?
After the last zero.
Where would you put the decimal to make this number be between 1 and 10?
Between the 2 and the 1

31. 2.10,000,000,000,000,000,000,000.
How many decimal places did you move the decimal? 23
When the original number is more than 1, the exponent is positive.
The answer in scientific notation is
2.1 x 1023

32. 1) Express 0.0000000902 in scientific notation.
Where would the decimal go to make the number be between 1 and 10?
9.02
The decimal was moved how many places? 8
When the original number is less than 1, the exponent is negative.
9.02 x 10-8

33. Write 28750.9 in scientific notation.
x 10-5
x 10-4
x 104
x 105

34. 2) Express 1.8 x 10-4 in decimal notation.
3) Express 4.58 x 106 in decimal notation.
4,580,000
On the graphing calculator, scientific notation is done with the button.
4.58 x 106 is typed

35. 4) Use a calculator to evaluate: 4.5 x 10-5 1.6 x 10-2
Type
You must include parentheses if you don’t use those buttons!!
(4.5 x ) (1.6 x )
Write in scientific notation.
x 10-3

36. 5) Use a calculator to evaluate:. 7. 2 x 10-9. 1
5) Use a calculator to evaluate: x x 102 On the calculator, the answer is:
6.E -11
The answer in scientific notation is
6 x
The answer in decimal notation is

37. 6) Use a calculator to evaluate (0. 0042)(330,000)
6) Use a calculator to evaluate (0.0042)(330,000). On the calculator, the answer is
1386.
The answer in decimal notation is
1386
The answer in scientific notation is
1.386 x 103

38. 7) Use a calculator to evaluate (3,600,000,000)(23)
7) Use a calculator to evaluate (3,600,000,000)(23). On the calculator, the answer is:
8.28 E +10
The answer in scientific notation is
8.28 x 10 10
The answer in decimal notation is
82,800,000,000

39. Write (2.8 x 103)(5.1 x 10-7) in scientific notation.

40. Write in PROPER scientific notation
Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 8) x 109
2.346 x 1011
9) x 104
on calculator: 642
6.42 x 10 2

41. Write 531.42 x 105 in scientific notation.

42. 6. 4 Objectives The student will be able to:
1. find the degree of a polynomial.
2. arrange the terms of a polynomial in ascending or descending order.
SOL: none
Designed by Skip Tyler, Varina High School

43. What does each prefix mean?
mono
one bi
two
tri
three

44. What about poly? one or more
A polynomial is a monomial or a sum/difference of monomials.
Important Note!!
An expression is not a polynomial if there is a variable in the denominator.

45. State whether each expression is a polynomial. If it is, identify it.
1) 7y - 3x + 4
trinomial
2) 10x3yz2
monomial 3)
not a polynomial

46. Which polynomial is represented byX 1 X 1 X2 X
x2 + x + 1
x2 + x + 2
x2 + 2x + 2
x2 + 3x + 2
I’ve got no idea!

47. The degree of a monomial is the sum of the exponents of the variables
The degree of a monomial is the sum of the exponents of the variables. Find the degree of each monomial.
1) 5x2 2
4a4b3c 8 -3

48. Which is biggest? 2 is the degree!
To find the degree of a polynomial, find the largest degree of the terms.
1) 8x2 - 2x + 7
Degrees:
Which is biggest? 2 is the degree!
2) y7 + 6y4 + 3x4m4
Degrees:
8 is the degree!

49. Find the degree of x5 – x3y2 + 42 3 5 10

50. A polynomial is normally put in ascending or descending order.
What is ascending order?
Going from small to big exponents.
What is descending order?
Going from big to small exponents.

51. Put in descending order:
8x - 3x2 + x4 - 4
x4 - 3x2 + 8x - 4
2) Put in descending order in terms of x:
12x2y3 - 6x3y2 + 3y - 2x
-6x3y2 + 12x2y3 - 2x + 3y

52. 3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x
-2x + 3y - 6x3y2 + 12x2y3
Put in ascending order:
5a a - a2
-3 + 2a - a2 + 5a3

53. Write in ascending order in terms of y: x4 – x3y2 + 4xy – 2x2y3
– 2x2y3 – x3y2 + 4xy + x4
x4 – x3y2– 2x2y3 + 4xy
4xy – 2x2y3 – x3y2 + x4

54. 6. 5 Objectives The student will be able to:
add and subtract polynomials.

55. 1. Add the following polynomials: (9y - 7x + 15a) + (-3y + 8x - 8a)
Group your like terms.
9y - 3y - 7x + 8x + 15a - 8a
6y + x + 7a

56. 2. Add the following polynomials: (3a2 + 3ab - b2) + (4ab + 6b2)
Combine your like terms.
3a2 + 3ab + 4ab - b2 + 6b2
3a2 + 7ab + 5b2

57. Add the polynomials. + x2 + 3x + 7y + xy + 8 x2 + 4y + 2x + 31 1 Y 1 1 1 Y 1 1 1 Y
x2 + 3x + 7y + xy + 8
x2 + 4y + 2x + 3
3x + 7y + 8
x2 + 11xy + 8

58. 3. Add the following polynomials using column form: (4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)
Line up your like terms.
4x2 - 2xy + 3y2
+ -3x2 - xy + 2y2
_________________________
x2 - 3xy + 5y2

59. Rewrite subtraction as adding the opposite.
4. Subtract the following polynomials: (9y - 7x + 15a) - (-3y + 8x - 8a)
Rewrite subtraction as adding the opposite.
(9y - 7x + 15a) + (+ 3y - 8x + 8a)
Group the like terms.
9y + 3y - 7x - 8x + 15a + 8a
12y - 15x + 23a

60. 5. Subtract the following polynomials: (7a - 10b) - (3a + 4b)
Rewrite subtraction as adding the opposite.
(7a - 10b) + (- 3a - 4b)
Group the like terms.
7a - 3a - 10b - 4b
4a - 14b

61. Line up your like terms and add the opposite.
6. Subtract the following polynomials using column form: (4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)
Line up your like terms and add the opposite.
4x2 - 2xy + 3y2
+ (+ 3x2 + xy - 2y2)
7x2 - xy + y2

62. Find the sum or difference. (5a – 3b) + (2a + 6b)

63. Find the sum or difference. (5a – 3b) – (2a + 6b)

64. 6.6 Objectives The student will be able to:
1. multiply a monomial and a polynomial.

65. Review: When multiplying variables,
add the exponents!
1) Simplify: 5(7n - 2)
Use the distributive property.
5 • 7n
35n - 10
- 5 • 2

66. 2) Simplify: 3) Simplify: 6rs(r2s - 3) 6a2 + 9a 6rs • r2s 6r3s2 - 18rs

67. 5) Simplify: - 4m3(-3m - 6n + 4p)
4) Simplify: 4t2(3t2 + 2t - 5)
12t4
5) Simplify: - 4m3(-3m - 6n + 4p)
12m4
+ 8t3
- 20t2
+ 24m3n
- 16m3p

68. 6) Simplify: (27x2 - 6x + 12) 16x3 - 28x2 + 4x
Fooled ya, didn’t I?!? Ha! Ha!
Here’s the real answer!
-9x3 + 2x2 - 4x

69. Simplify 4y(3y2 – 1)
7y2 – 1
12y2 – 1
12y3 – 1
12y3 – 4y

70. Simplify -3x2y3(y2 – x2 + 2xy) -3x2y5 + 3x4y3 – 6x3y4

71. 6.7 Objective The student will be able to:
multiply two polynomials using the FOIL method, Box method and the distributive property.

72. There are three techniques you can use for multiplying polynomials.
The best part about it is that they are all the same! Huh? Whaddaya mean?
It’s all about how you write it…Here they are!
Distributive Property
FOIL
Box Method
Sit back, relax (but make sure to write this down), and I’ll show ya!

73. 1) Multiply. (2x + 3)(5x + 8)
Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8).
10x2 + 16x + 15x + 24
Combine like terms.
10x2 + 31x + 24
A shortcut of the distributive property is called the FOIL method.

74. The FOIL method is ONLY used when you multiply 2 binomials
The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. 2) Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

75. (y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial.

76. (y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial.

77. (y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial.
y2 + 7y + 3y

78. (y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial.
y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21

79. Remember, FOIL reminds you to multiply the:
First terms
Outer terms
Inner terms
Last terms

80. The third method is the Box Method. This method works for every problem!
Here’s how you do it. Multiply (3x – 5)(5x + 2)
Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.
This will be modeled in the next problem along with FOIL. 3x -5 5x +2

81. 3) Multiply (3x - 5)(5x + 2) 3x -5 5x +2 15x2 +6x -25x 15x2 -25x -10
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
15x2 - 19x – 10 3x -5 5x +2
+6x
-25x
15x2
-25x
-10
+6x
-10
You have 3 techniques. Pick the one you like the best!

82. 4) Multiply (7p - 2)(3p - 4) 7p -2 3p -4 21p2 -28p -6p 21p2 -6p +8
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
21p2 – 34p + 8 7p -2 3p -4
-28p
-6p
21p2
-6p +8
-28p +8

83. Multiply (y + 4)(y – 3) y2 + y – 12 y2 – y – 12 y2 + 7y – 12

84. Multiply (2a – 3b)(2a + 4b) 4a2 + 14ab – 12b2 4a2 – 14ab – 12b2
4a2 + 8ab – 6ba – 12b2
4a2 + 2ab – 12b2
4a2 – 2ab – 12b2

85. Group and combine like terms.
5) Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not BOTH binomials. You must use the distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20

86. 5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2
-5x +4 2x -5
2x3
-10x2
+8x
Almost done!
Go to the next slide!
-5x2
+25x
-20

87. 5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!+4 2x -5
2x3
-10x2
+8x
-5x2
+25x
-20
2x3 – 15x2 + 33x - 20

88. Multiply (2p + 1)(p2 – 3p + 4) 2p3 + 2p3 + p + 4 y2 – y – 12