Transcript Multiply/Divide Monomials and Polynomials

Topic: How do we divide
monomials? 35x y
4 10
7 x3 y 2
Three things everyone should know
about exponents.
1) Parentheses matter: 82   8 2
2) Anything to the zero power equals 1.
x0 = 1
Well almost anything
3) Negative exponents do not negate 52  25
the base they move it! 2 1
5 
52
Three things everyone should know
about exponents.
1) Parentheses matter: 82   8 2
2) Anything to the zero power equals 1.
x0 = 1
Well almost anything
3) Negative exponents do not negate 52  25
2
the base they move it! 5 x 2
8
 
 
   
 8 
 5x 
1. Be able to divide polynomials
2. Be able to simplify expressions involving powers of
monomials by applying the division properties of powers.
3
4x y
2 xy
3
 3xy 

3 
 yx 
2
5
8 2
36a b
ab
Monomial: A number, a variable, or the product of a number
and one or more variables
Constant: A monomial that is a real number.
Power: An expression in the form xn.
Base: In an expression of the form xn, the base is x.
Exponent: In an expression of the form xn, the exponent is n.
Quotient: The number resulting by the division of one number by another.
Product of Powers: When two numbers with the same base are
multiplied together, add the exponents and leave the base unchanged.
a a  a
m
n
m n
Power of a Product: In a product raised to a power, the exponent
applies to each factor of the product.
3xy
2
3 x y
2
2
2
Power of a Power: When a power is raised to another power,
multiply the exponents and leave the base unchanged.
a 
m n
a
m n
Remember: Follow the order of operations when applying more than
one property!
3x
2
 
3
 x y  3x
3
23

3
 
y 3 x
3
5 3
15 3
53 3

27
x
y

27
x
y
y
3
3
3
x
Simplify:
x2
Step 1: Rewrite the expression in expanded form
3x 3 3  x  x  x

2
x
x x
Step 2: Simplify.
3 x  x  x
xx
 3x
Remember: A number divided by itself is 1.
Dividing Monomials
 For all real numbers a, and integers m and n:
m
a
mn

a
n
a
Dividing Monomials
Ex 1: Simplify using positive exponents only
3 5
20m n
5

5mn
2
4m
Dividing Monomials
Ex 2: Simplify using positive exponents only
4
x
5x y

1 8
3
45 x y
9y
3
5
Dividing Monomials
Ex 3: Simplify using positive exponents only
3
30 x y
7 5
6x y
5
5
 10
x
 3x 

 y
2
Simplify: 
Step 1: Write the exponent in expanded form.
2
 3x  3x 3x
   
y y
 y
For all real numbers a and b, and
integer m:
Step 2: Multiply and simplify.
m
9x
3x 3x 3 x  3 x 3 x
 2  2
 
y
y
y y
y y
2
2
2
a
a
   m
b
b
m
Fractions to a Power
For all real numbers a and b, and
integer m:
m
a
a
   m
b
b
m
Ex 4: Simplify using positive exponents only
 2a b 


 a 
2
3
 2a b 


 a 
2
 2ab 
3
23 a 3b3
8a 3b 3
3
Ex 5: Simplify using positive exponents only
 2a b 
 7 
 ab 
2
3
 4x y

 2 xy
3
 4x y

 2 xy
3
3
3
  2
   
  xy 
2
2
  2   4 31 31  2 2
       x  y   2 2
 x y
  xy   2
4
 2x2 y 2  2 2
x y


2x2 y 2  4

x2 y2
 42 x
8
2 2
THINK!
y
2 2
x2-2 = x0 = 1