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