Transcript 9.1 Multiplying and Dividing Rational Expressions

9.1 Multiplying and Dividing
Rational Expressions
Complex fractions
Definition of a Rational Expression
In mathematics, a rational function is any
function which can be written as the ratio
of two polynomial functions.
Since they are fractions
Simplify a Rational Expression.
3 y y  7
2
 y  7 y  9


Divide the same expression from numerator
and denominator
Since they are fractions
Simplify a Rational Expression.
3 y y  7
2
 y  7 y  9


Divide the same expression from numerator
and denominator
3 y y  7
3y
 2
2
 y  7 y  9 y  9


Since they are fractions
There are numbers that will not work in the
Rational expression.
3 y y  7
3y
 2
2
 y  7 y  9 y  9


They are -7, 3 and -3. Why?
Since they are fractions
Factor out the denominator.
3 y y  7
 y  7 y  3 y  3
Can you see why -7, 3 and -3 do not work
as answers?
Since they are fractions
Factor out the denominator.
3 y y  7
 y  7 y  3 y  3
-7, 3 and -3 do not work as answers
because they make the denominator zero.
They are called Excluded Values.
Simplify
Assume the denominator cannot equal zero.
a b  2a
3
3
2a  a b
4
4
Simplify
Assume the denominator cannot equal zero.
a b  2a
a b  2

3
3
3
2a  a b a 2  b
4
4
4
Factor out a negative one to move things
around
Simplify
Assume the denominator cannot equal zero.
a b  2a
a b  2 
 3
3
3
2a  a b a 2  b 
4
4
 a 2  b 

3
a 2  b 
4
4
Simplify
Assume the denominator cannot equal zero.
a b  2a
a b  2 
 3
3
3
2a  a b a 2  b 
4
4
4
 a 2  b   a
 3  a
3
a 2  b 
a
4
4
Multiply the fractions
Reduce to simplest form
12 5

7 3
Multiply the fractions
Reduce to simplest form.
Multiply straight across
12 5 60
 
7 3 21
Multiply the fractions
Reduce to simplest form
12 5 60 3 20
 
 
7 3 21 3 7
Multiply the fractions
Reduce to simplest form.
12 5 60 3 20
 
 
7 3 21 3 7
What happenedhere
4 5 20
 
7 1 7
Multiply the fractions
Reduce before multiply.
2
8x 7 y

3
3
21y 16x
Multiply the fractions
Reduce before multiply.
2
8x 7 y

3
3
21y 16x
1 1
1
 2  2
3 y 2x
6x y
Multiply the fractions
Reduce before multiply.
4
2
5a c 24bc

3 2
12b 15a b
Multiply the fractions
Reduce before multiply.
4
2
5a c 24bc

3 2
12b 15a b
2
3
ac 2c
2ac
 2 
2
1 3b
3b
Multiply the fractions
Reduce before multiply.
k 3
1 k
 2
k  1 k  4k  3
2
Multiply the fractions
Reduce before multiply.
k 3
1 k 2
 2
k  1 k  4k  3
k  3   k 2  1
k  1 k  3k  1
Multiply the fractions
Reduce before multiply.
k 3
1 k
 2
k  1 k  4k  3
2
k  3   k 2  1
k  1 k  3k  1
k  3   1k  1k  1  1   1  1
k  1 k  3k  1 1 1
What is the difference of Division
and Multiply of fractions?
Do you remember?
2 3

7 4
What is the difference of Division
and Multiply of fractions?
When dividing you multiply by the inverse of
the second fraction.
2 3

7 4
2 4 8
 
7 3 21
Divide the Rational Expressions
You can only Reduce when Multiplying
2
10 px
5 px

2
2 2
3c d 6c d
Divide the Rational Expressions
You can only Reduce when Multiplying
2
10 px
5 px

2
2 2
3c d 6c d
2
2
2
10 px 6c d
2 x 2d



 4dx
2
3c d 5 px
1 1
Definition of Complex fraction
A fraction with fractions in the numerator or
denominator.
1
5
3
7
How to simplify a Complex fraction
Remember fractions are division statements.
1
3  15 17  7
5
3 7
3 5 15
7
Simplify the Complex fraction
Remember fractions are division statements.


x
 2

2 
 9x  4 y 
3
 x



 2 y  3x 
2
Simplify the Complex fraction
Remember fractions are division statements.


x
 2

2 
2
3
x
x
 9x  4 y  

2
2
3
9x  4 y
2 y  3x
 x



 2 y  3x 
2
Simplify the Complex fraction
x
x
x
2 y  3x

 2

2
2
2
3
9x  4 y
2 y  3x
9x  4 y
x
2
3
2
Simplify the Complex fraction
x
x
x
2 y  3x

 2

2
2
2
3
9x  4 y
2 y  3x
9x  4 y
x
2
3
2
x
 13x  2 y 

3
3x  2 y 3x  2 y 
x
2
Simplify the Complex fraction
x
x
x
2 y  3x

 2

2
2
2
3
9x  4 y
2 y  3x
9x  4 y
x
2
3
2
x
 13x  2 y 

3
3x  2 y 3x  2 y 
x
2
1
1
1


3x  2 y  x x3x  2 y 
Homework
Page 476 – 477
# 15,19, 23,
27, 31, 35,
37, 41
Homework
Page 476 – 477
# 16,20, 24,
28, 32, 36,
38, 40