This document provides instructions for simplifying rational expressions through multiplying and dividing rational expressions. It explains the steps to multiply rational expressions by multiplying the numerators and denominators. It also explains how to divide rational expressions by multiplying the original expression by the reciprocal of the divisor. Examples are provided to demonstrate multiplying and dividing rational expressions as well as factoring expressions to determine values that make them undefined.

1. Simplifying
Multiplying
Dividing
Grade 9 B – T3 -W 5- 13/14

2. Simplifying

3. Remember, denominators
can not = 0.
Now,lets go through the steps to
simplify a rational expression.
Examples of rational expressions
2
4 8 4 7
, ,
3 3 5 9
x y
x x y y
 
  

4. Simplify:
7x  7
x2
1
Step 1: Factor the numerator and
the denominator completely looking
for common factors.
7x  7  7(x 1)
x2
1  (x 1)(x 1)
Next

5. 7x  7
x2
1

7(x 1)
(x 1)(x 1)
What is the common factor?
x1
Step 2: Divide the numerator and
denominator by the common factor.

6. 7(x 1)
(x  1)(x 1)

7(x 1)
(x  1)(x 1)
1
1
Step 3: Multiply to get your answer.
Answer:
7
x 1

7. Looking at the answer from the
previous example, what value of x
would make the denominator 0?
x= -1
The expression is undefined when the
values make the denominator equal to 0

8. How do I find the values
that make an expression
undefined?
Completely factor the original
denominator.

9. Ex:
2ab(a  2)(b  3)
3ab(a2
 4)
3ab(a2
 4)  3ab(a  2)(a  2)
The expression is undefined when:
a= 0, 2, and -2 and b= 0.
Factor the denominator

10. Lets go through another example.
3a3
 a4
2a
3
 6a
2
3a3
 a4
2a
3
 6a
2 
a3
(3 a)
2a
2
(a 3)
Factor
out the
GCF
Next

11. 3
2
2 ( 3)
(3 )a
a a
a


  3 factored is 1( 3)a a  
cancel like factors
3
2
1 ( 3)
2 ( 3)
a a
a a
 


1
1
3
2
1( 3)
2 ( 3)
a
a a
a

 


12. 3
2
1
2
a
a
 2
cancel out the like factor a
1
2
a
1
a
answer
What values is the original expression undefined?

13. Now try to do some on your own.
2
2
3 2
3 2
5 6
1)
9
5 10
2)
6 16
x x
x
x x
x x x
 


 
Also find the values that make
each expression undefined?

14. Multiplying and
Dividing

15. Multiplying and Dividing
Rational Expressions
Remember that a rational number
can be expressed as a quotient of
two integers. A rational expression
can be expressed as a quotient of
two polynomials.

16. Remember how to multiply fractions:
First you multiply the numerators
then multiply the denominators.
5 2
:
6 20
Ex 
10 1
120 12

5 2
6 20


17. The same method can be used
to multiply rational expressions.
Ex:
4a2
5ab
3 
3bc
12a
3 
4  a  a 3  b c
5 a  b  b b 12  a  a a
11 1 1 1
1 1 1 1

c
5b2
a2

18. Let’s do another one.
Ex:
x3
 3x2
x
2
 5x  6

x2
10x  9
x
2
 6x  27
Step #1: Factor the numerator
and the denominator.
x2
(x  3)
(x  6)(x 1)

(x 1)(x  9)
(x  9)(x 3)
Next

19. Step #2: Divide the numerator and
denominator by the common factors.
x2
(x  3)
(x  6)(x 1)

(x 1)(x  9)
(x  9)(x 3)
1
1
1
1
1
1

20. Step #3: Multiply the numerator
and the denominator.
x2
x  6
Remember how to divide fractions?

21. Multiply by the reciprocal of the
divisor. 4
5

16
25

4
5

25
16

4  25
516
1
1
5
4

5
4

22. Dividing rational expressions uses
the same procedure.
Ex: Simplify
y  2
y
2
10y  24

y2
 2y
y
2
 2y  8

23. y  2
y
2
10 y  24

y2
 2y
y
2
 2y  8

y  2
y
2
10y  24

y2
 2y  8
y
2
 2y

y  2
(y 12)(y  2)

(y  4)(y  2)
y(y  2)

1 1
1 1
Next
4
( 12)
y
y y



24. Now you try to simplify the
expression:
x  3
x
2
 4x 12

2x2
 6x
x  2

25. Choose the correct answer .

26. Answer:
1
2x(x  6)
Now try these on your own.
1)
x + 3
2x3
 2x2

x2
 7x  6
x2
10x  21
2)
3x  6
7x  7

5x 10
14x 14