This document discusses properties and rules for operations involving polynomials. It defines key terms like simplify and degree of a polynomial. It then lists properties for operations with powers such as product of powers, power of a power, power of a product, and quotient of powers. Examples are provided to illustrate each property rule. Finally, it covers properties for operations with zero powers, negative exponents, and quotients of powers.

1. Section 4-1
Operations with Polynomials

2. Essential Questions
• How do you multiply, divide, and simplify monomials
and expressions involving powers?
• How do you add, subtract, and multiply polynomials?

3. Vocabulary
1. Simplify:
2. Degree of a Polynomial:

4. Vocabulary
1. Simplify: To eliminate all powers by applying all
rules so that there are no parentheses or
negative exponents remaining
2. Degree of a Polynomial:

5. Vocabulary
1. Simplify: To eliminate all powers by applying all
rules so that there are no parentheses or
negative exponents remaining
2. Degree of a Polynomial: The sum of all of the
powers of the term with the highest degree

6. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers

7. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n

8. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2

9. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
x5

10. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5

11. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29

12. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512

13. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn

14. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3

15. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21

16. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4

17. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38

18. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561

19. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm

20. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3

21. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3

22. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3

23. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3
(3x4
)2

24. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3
(3x4
)2
9x8

25. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3
(3x4
)2
9x8
xm
xn
= xm−n

26. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3
(3x4
)2
9x8
xm
xn
= xm−n x5
x2
= x3

27. Properties of Powers
Property Rule Example 1 Example 2
Product of
Powers
Power of a
Power
Power of a
Product
Quotient of
Powers
xm
i xn
= xm+n x3
i x2
24
i 25
x5
29
512
(xm
)n
= xmn (x7
)3
x21
(32
)4
38
6561
(ax)m
= am
xm (2x)3
23
x3
8x3
(3x4
)2
9x8
xm
xn
= xm−n x5
x2
= x3 6x12
y4
8x5
y
=
3x7
y3
4

28. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent

29. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym

30. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8

31. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10

32. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1

33. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1 (x7
)0
= 1

34. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1 (x7
)0
= 1 3x12
y4
z11
5x4
yz9
⎛
⎝
⎜
⎞
⎠
⎟
0
= 1

35. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1 (x7
)0
= 1 3x12
y4
z11
5x4
yz9
⎛
⎝
⎜
⎞
⎠
⎟
0
= 1
x−n
=
1
xn

36. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1 (x7
)0
= 1 3x12
y4
z11
5x4
yz9
⎛
⎝
⎜
⎞
⎠
⎟
0
= 1
x−n
=
1
xn
1
x−n
= xn

37. Properties of Powers
Property Rule Example 1 Example 2
Power of a
Quotient
Zero
Power
Negative
Exponent
x
y
⎛
⎝
⎜
⎞
⎠
⎟
m
=
xm
ym
x3
y2
⎛
⎝
⎜
⎞
⎠
⎟
4
=
x12
y8
2x4
3y2
⎛
⎝
⎜
⎞
⎠
⎟
5
=
32x20
243y10
x0
= 1 (x7
)0
= 1 3x12
y4
z11
5x4
yz9
⎛
⎝
⎜
⎞
⎠
⎟
0
= 1
x−n
=
1
xn
1
x−n
= xn x−4
y
z−5
=
yz5
x4

38. Example 1
Simplify.
a. (a−3
)(a2
b4
)(c−1
) b.
n2
n10
c.
3a2
b4
⎛
⎝⎜
⎞
⎠⎟
2

39. Example 1
Simplify.
a. (a−3
)(a2
b4
)(c−1
) b.
n2
n10
a−1
b4
c−1
c.
3a2
b4
⎛
⎝⎜
⎞
⎠⎟
2

40. Example 1
Simplify.
a. (a−3
)(a2
b4
)(c−1
) b.
n2
n10
a−1
b4
c−1
b4
ac
c.
3a2
b4
⎛
⎝⎜
⎞
⎠⎟
2

41. Example 1
Simplify.
a. (a−3
)(a2
b4
)(c−1
) b.
n2
n10
a−1
b4
c−1
b4
ac
1
n8
c.
3a2
b4
⎛
⎝⎜
⎞
⎠⎟
2

42. Example 1
Simplify.
a. (a−3
)(a2
b4
)(c−1
) b.
n2
n10
a−1
b4
c−1
b4
ac
1
n8
c.
3a2
b4
⎛
⎝⎜
⎞
⎠⎟
2
9a4
b8

43. Example 2
Determine whether each expression is a polynomial.
If is is, state the degree of the polynomial.
a. c4
− 4 c +18 b. −16p5
+
3
4
p2
t7
c. x 2
− 3x −1
+ 7

44. Example 2
Determine whether each expression is a polynomial.
If is is, state the degree of the polynomial.
a. c4
− 4 c +18 b. −16p5
+
3
4
p2
t7
c. x 2
− 3x −1
+ 7
No

45. Example 2
Determine whether each expression is a polynomial.
If is is, state the degree of the polynomial.
a. c4
− 4 c +18 b. −16p5
+
3
4
p2
t7
c. x 2
− 3x −1
+ 7
No Yes

46. Example 2
Determine whether each expression is a polynomial.
If is is, state the degree of the polynomial.
a. c4
− 4 c +18 b. −16p5
+
3
4
p2
t7
c. x 2
− 3x −1
+ 7
No Yes
Degree: 9

47. Example 2
Determine whether each expression is a polynomial.
If is is, state the degree of the polynomial.
a. c4
− 4 c +18 b. −16p5
+
3
4
p2
t7
c. x 2
− 3x −1
+ 7
No Yes
Degree: 9
No

48. Example 3
Simplify each expression.
a. (2a3
+ 5a − 7)− (a3
− 3a + 2)
b. (4x 2
− 9x + 3)+ (−2x 2
− 5x − 6)

49. Example 3
Simplify each expression.
a. (2a3
+ 5a − 7)− (a3
− 3a + 2)
2a3
+ 5a − 7 − a3
+ 3a − 2
b. (4x 2
− 9x + 3)+ (−2x 2
− 5x − 6)

50. Example 3
Simplify each expression.
a. (2a3
+ 5a − 7)− (a3
− 3a + 2)
2a3
+ 5a − 7 − a3
+ 3a − 2
a3
+ 8a − 9
b. (4x 2
− 9x + 3)+ (−2x 2
− 5x − 6)

51. Example 3
Simplify each expression.
a. (2a3
+ 5a − 7)− (a3
− 3a + 2)
2a3
+ 5a − 7 − a3
+ 3a − 2
a3
+ 8a − 9
b. (4x 2
− 9x + 3)+ (−2x 2
− 5x − 6)
4x 2
− 9x + 3 − 2x 2
− 5x − 6

52. Example 3
Simplify each expression.
a. (2a3
+ 5a − 7)− (a3
− 3a + 2)
2a3
+ 5a − 7 − a3
+ 3a − 2
a3
+ 8a − 9
b. (4x 2
− 9x + 3)+ (−2x 2
− 5x − 6)
4x 2
− 9x + 3 − 2x 2
− 5x − 6
2x 2
−14x − 3

53. Example 4
Find − y (4y 2
+ 2y − 3).

54. Example 4
Find − y (4y 2
+ 2y − 3).
−4y 3
− 2y 2
+ 3y

55. Example 5
0.001x 2
+ 5x + 500
Matt Mitarnowski estimates that the cost in dollars
associated with selling x units Shecky’s Shoe Shine
is given by the expression . The
revenue from selling x units is given by 10x. Write a
polynomial to represent the profits generated by the
product if profit = revenue − cost.
x

56. Example 5
0.001x 2
+ 5x + 500
Matt Mitarnowski estimates that the cost in dollars
associated with selling x units Shecky’s Shoe Shine
is given by the expression . The
revenue from selling x units is given by 10x. Write a
polynomial to represent the profits generated by the
product if profit = revenue − cost.
x
profit = 10x − (0.001x 2
+ 5x + 500)

57. Example 5
0.001x 2
+ 5x + 500
Matt Mitarnowski estimates that the cost in dollars
associated with selling x units Shecky’s Shoe Shine
is given by the expression . The
revenue from selling x units is given by 10x. Write a
polynomial to represent the profits generated by the
product if profit = revenue − cost.
x
profit = 10x − (0.001x 2
+ 5x + 500)
profit = −0.001x 2
+ 5x − 500

58. Example 6
Find (a2
+ 3a − 4)(a + 2)

59. Example 6
Find (a2
+ 3a − 4)(a + 2)
a3
+ 2a2
+ 3a2
+ 6a − 4a − 8

60. Example 6
Find (a2
+ 3a − 4)(a + 2)
a3
+ 2a2
+ 3a2
+ 6a − 4a − 8
a3
+ 5a2
+ 2a − 8