Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 1 of 4)
Definition
1)
Monomial—What is a monomial?
The following are monomials:
So monomials can include:
5
3
2
5
3
2x
– 8.5
y2
 8.5 xy 2
9a 2 b 3 c 7
0
5
However, these are not monomials:
2+x
x 2  8. 5
9
x
x
So a monomial cannot have:
Vocabulary
Using the monomials above, identify examples of the following definitions.
2)
A ___________________ is a monomial with no variables,
Examples:
3)
In a monomial, the ___________________ is the numeric factor of the variable (or variables)
Examples:
4)
The _____________________ of a monomial is the sum of the exponents on the variables only
Monomial
Examples:
2x
5
3
Monomial
1
y2
5
 8.5 xy 2
©2010, TESCCC
9a 2 b 3 c 7
0
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 2 of 4)
Simplifying Monomials
Multiplying
Sample:
Expand:
Re-order:
Dividing
2
3
4
(5 x y )( 4 xy )
5x x y y y 4x y y y y
54x x x y y y y y y y
Sample:
Expand:
Simplify:
18 x 4 y 3
6 xy 2
18  x  x  x  x  y  y  y
6 x
y y
Simplify:


What’s the short-cut?
the coefficients
What’s the short-cut?
the coefficients
the exponents

the exponents

What’s the rule?
What’s the rule?
am

an
Note: Bases must be the same.
am  an 
Note: Bases must be the same.
Other Rules
a n 
a 
m n
Samples
1
an
 a mn
m
am
a
   m
b
b
(ab ) m  a m b m
©2010, TESCCC
5
2

(5 2 ) 3 

5 2
8
x =
b3
=
b9
( x 4 )5 =
(b 2 ) 6 =
4
7
x
  =
y

(5 x 2 )2 
( xy 2 ) 6 
08/01/10
 b3
 2
c
5

 =

( 4 x 2 y 4 )3 
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 3 of 4)
Sample Problems
1)
 2ab  4a b c 
2)
3x 
3)
x11
x5
4)
 6x y   xyz 
5)
 6a
6)
 3x 3 y 6 
 5 2 1 
x y z 
7)
A rectangle has a width represented by 3 x 2 y 6 and a length represented by 8 x 5 y 3 . What
expression can be used to represent the area of the rectangle?
2
3
b2 
3
3
3
2 3
2
3
3
2
8x 5 y 3
3x 2 y 6
8)
The area of the triangle below is represented by 14 x 4 y 9 . Find the expression that represents
the base of the triangle.
7x 2y 5
b=?
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 4 of 4)
Practice Problems
Simplify the following expressions.
1)
x3  x3  x  x5
2)
 3xy 
3)
 x6y 6
x3y 4
4)
3 x
5)
ab 4 c 6
a 5 bc 2
6)
2a 5 b 3 c 3
8a 3 b 3 c
7)
40a 1b 7
20a 5 b 3
8)
 15m n m n 
9)
2 3
2

y 6  4x 2 y 6
5
8
3

2 2
45m 4 n
The height of a triangle is represented by the expression 15 p 6 qr 3 . The base is represented by
8 p 2 q 3 r 5 . Find the expression that can be used to represent the area of the triangle.
w=?
10) The length and area of a rectangle are given in the diagram below. Find the expression that can
be used to represent the width of the rectangle.
Area = 72m15n10
4m3n7
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 1 of 4) KEY
Definition
1)
Monomial—What is a monomial?
The following are monomials:
5
3
2
5
3
2x
– 8.5
y2
 8.5 xy 2
9a 2 b 3 c 7
0
So monomials can include:
 Numbers
 Variables
 Products of numbers and variables
5
However, these are not monomials:
2+x
x 2  8. 5
9
x
x
So a monomial cannot have:
 Variables under a radical (square root)
 Variables in a denominator
 Addition or subtraction
Vocabulary
Using the monomials above, identify examples of the following definitions.
2)
A constant is a monomial with no variables,
Examples: 2, 0, – 8.5,
3)
5
3
,
5
In a monomial, the coefficient is the numeric factor of the variable (or variables)
2
Examples: 2 (in 2x), -8.5 (in 8.5 xy ),
4)
5
3
(in
5
3
y 2 ), 9 (in 9a 2b3c 7 )
The degree of a monomial is the sum of the exponents on the variables only
Examples:
Monomial
Degree
Monomial
Degree
2x
1
9a 2b3c 7
12
y2
2
5
0
3
0
0
5
3
 8.5 xy 2
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 2 of 4) KEY
Simplifying Monomials
Multiplying
Sample:
Expand:
Re-order:
Dividing
2
3
4
(5 x y )( 4 xy )
5x x y y y 4x y y y y
54x x x y y y y y y y
Sample:
Expand:
20x3y7
Simplify:
18 x 4 y 3
6 xy 2
18  x  x  x  x  y  y  y
6 x
y y
Simplify: 3x3y
 What’s the short-cut?
Multiply
the coefficients
 What’s the short-cut?
Divide
the coefficients
Add
Subtract

the exponents

What’s the rule?
am  an 
What’s the rule?
am
a m –n

n
a
Note: Bases must be the same.
a m+n
Note: Bases must be the same.
Other Rules
an 
a 
m n
Samples
1
an
 a mn
m
am
a

b
bm
 
 ab 
m
©2010, TESCCC
the exponents
 ambm
1
25
4
x =
1
x4
b3
=
b9
1
b6
(5 2 ) 3 
(5)6
(x 4 ) 5 =
x20
(b 2 ) 6 =
b-12

25
64
x
  =
y
x7
y7
 b3
 2
c
25x4
( xy 2 ) 6 
x6 y12
5
2
5 2
8


(5 x 2 )2 
7
08/01/10
5

 =

( 4 x 2 y 4 )3 
b15
c 10
64x6 y12
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 3 of 4) KEY
Sample Problems
1)
2ab  4a b c 
2
3
3
2)
4 5
-8a b c
3x 
2 3
27x6
6 x

4)
y 3 xyz 
6x5y6z3
6)
  3x 3 y 6 
 5 2 1 
x y z 
9y16 z2
x4
3)
x 11
x5
x6
5)
6a
7)
A rectangle has a width represented by 3 x 2 y 6 and a length represented by 8 x 5 y 3 . What
expression can be used to represent the area of the rectangle?
A = length  width
A = (3x2y6)(8x5y3)
A = 24x7y9
3

b2
a9
216b6
3
2
3
2
8x 5 y 3
3x 2 y 6
8)
The area of the triangle below is represented by 14 x 4 y 9 . Find the expression that represents
the base of the triangle.
A = ½ bh
14x4y9 = ½ b(7x2y5)
28x4y9 = b(7x2y5)
2 5
28x4 y9
7x y
b=
7x2 y5
2 4
b = 4x y
b=?
©2010, TESCCC
08/01/10
Algebra 2
HS Mathematics
Unit: 06 Lesson: 01
Investigating Monomials (pp. 4 of 4) KEY
Practice Problems
1)
Simplify the following expressions.
 3xy 
x3  x3  x  x5
x12
2)
 x6y 6
x3y 4
-x3y2
4)
5)
ab 4 c 6
a 5 bc 2
b3 c4
a4
6)
2a 5 b 3 c 3
8a 3 b 3 c
a2 c2
4
7)
40a 1b 7
20a 5 b 3
2a4
b10
8)
 15m n m n 
3)
9)
2 3
-27x3 y6
3 x

y 6  4x 2 y 6
-12x y
2
4 12
5
8
3

2 2
45m 4 n
-n3
3m5
The height of a triangle is represented by the expression 15 p 6 qr 3 . The base is represented by
8 p 2 q 3 r 5 . Find the expression that can be used to represent the area of the triangle.
A = 60p8q4r8
w=?
10) The length and area of a rectangle are given in the diagram below. Find the expression that can
be used to represent the width of the rectangle. Width = 18m12n3
Area = 72m15n10
4m3n7
©2010, TESCCC
08/01/10