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Monomials
An expression that is either
a number,
a variable,
or a product of numerals and
variables with whole number
exponents.
Monomials
5x3y12 is a monomial
2y
is
not
a
monomial
2
x
1
3
5x y z is not a monomial
3
12
Vocabulary
Constants
monomials that contain no
variables
Example
3 or -22
Coefficient
Numeric factor of the term
-32x3y12z15 coefficient = -32
Vocabulary (continued)
Degree of a Monomial
The sum of the exponents of the
variables
3x4
degree = 4
-32x3y12z15
degree = 3+12+15
= 30
5
degree = 0
Vocabulary (continued)
Power
An expression in the form of xn
Can also refer to the exponent
Product of Powers
For any real number a and
integers m and n,
am · an =am+n
2 3 · 25
8
=2 · 2 · 2 · 2 · 2 · 2 · 2 · 2= 2
Quotient of Powers
For any real number a and
integers m and n,
a
m n
a
n
a
m
5 555 5 5
3

5
2
5
55
5
Quotient of Powers
Find the quotient
2
5
2
8
2
3
2
5
2
8
2
3
2 2 2
1
 5
22222222 2
NEGATIVE EXPONENTS
For any real number a≠0
and any integer n, a-n= 1n
a
2
3
1
1
 3 
2
8
1
8

b
b 8
Vocabulary (continued)
Simplify
rewrite expression
No parenthesis
No negative exponents
Multiply variables
Combine like terms
Simplify
(-2a3b)(-5ab4)
Multiply Coefficients
(-2)(-5)=10
Multiply Variables
(a3)(a) = a4
(b)(b4) = b5
10a4b5
Simplify
Try this one
14a b c
5
3
21a bc
3
2b c
2
3a
3
4
9
6
PROPERTIES OF POWERS
• Power of a Power: (am)n=amn
• Power of a Product: (ab)m=ambm
n
n
a
a


• Power of a Quotient:   
b 
a 
 
b 
bn
n
n
bn
b 
   n
a
a 
Properties of Powers
b   b
4
2
5
8
b
 2a  
2 5
 2 
b 
 b 

x
3
4
 2a 
3

 
x 
4
5
24
32a 5

b 10
34
 4
x
81
 4
x
Scientific Notation
FORM
a x 10n
1  a  10
n is an integer
Write in Scientific Notation
4,560,000
0.000092
Multiply Numbers in
Scientific Notation
(a x 10n) (b x 10m) = (ab x 10n+m)
Check and make sure 1  ab  10
(1.8 x 104) (4 x 107)
(5 x 103) (7 x 108)
Divide Numbers in
Scientific Notation
a  10m
m n

a

b

10
b  10n
1  a  b  10
Check and make sure
2.7  10
10
9  10
6
Polynomials
A monomial or a sum of monomials.
Monomial – a polynomial with exactly one term
Binomial – a polynomial with exactly two terms
Trinomial – a polynomial with exactly three terms
Polynomial Vocabulary
Term
Each monomial in a polynomial
Like Terms
Terms whose variable factors are
exactly the same
Degree of the Polynomial
The highest degree of its terms
Polynomials
• Indicate if the following is a polynomial,
• If so classify according to the number of
terms
• Indicate the degree of the polynomial
c  4 c  18
Not a polynomial
3 2 7
16 p  p q
4
Polynomial- Binomial- 9
4
5
Simplify
(2a3+5a-7) + (a3-3a+2)
3a3+2a-5
(3b3+2b2-4b+3) - (b3-2b2+3b-4)
2b3+4b2-7b+7
-3y(4y2+2y-3)
-12y3 - 6y2 + 9y
Polynomial Vocabulary
(continued)
Leading Term
The term with the highest degree
Leading Coefficient
The coefficient of the leading term
Descending Order
A polynomial is written in descending order
for the variable x when the term with the
greatest exponent for x is first, and each
subsequent term has an exponent for x less
than the prior term.
Example: Write the following in descending
order for the variable a.
4a4 + a2 - 7a3 +6a5 + 12a8 + 4
12a8 + 6a5 + 4a4 - 7a3 + a2 + 4
Multiplying Polynomials
2p  3 4p  1
8p  14 p  3
2
Multiplying Polynomials
a
2
 3a  4  2a  1 
2a  5a  11a  4
3
2