1. Rational Expressions and
Equations
PRE-CAL
MR. WATT
BY DANNY COLLICUTT, RUSTINE
PADILLA, JUNEBE CORONIA

2. Simplifying Rational Expressions
 We define a Rational Expression as a fraction where
the numerator and the denominator are
polynomials.
 Ex. x²-y²/ (x-y)²

3.  To simplify we first factor the polynomials then
cancel any common factors
 (x - y)(x + y) x +y
=
(x - y)2 x-y
(x-y)² is equal to (x-y)(x-y) so we can cancel out one of the (x-y)

4. Example
 Now try : 3x2 - 4x/ 2x2 - x
Click to see the answer and the steps used

5. Answer
 3x2 - 4x x(3x - 4) 3x - 4
= =
2x2 - x x(2x - 1) 2x - 1

6. Review of Addition and Subtraction of Fractions
3 5 8 (2)(4) 2
+ = = =
20 20 20 (4)(5) 5
 first combine the numerators since the denominators
are the same. Then factor both the numerator and
denominator and finally cross cancel.

7. Addition and Subtraction of Rational Expressions
 To add and subtract rational functions, we follow the
same method as fractions.

 Step 1 Factor everything and find the least common
denominator.
 Step 2 Multiply the numerators and the denominators
by the appropriate denominator so that the denominator
becomes the least common Denominator.
 Step 3 Add the numerators together.
 Step 4 Factor the numerator.
 Step 5 Cancel any common factors.

8. Example
3x + 1 x
First Factor everything and find the
+ lowest common denominator
x2 - 1 x+1
3x + 1 x
+
The LCD is (x - 1)(x + 1)
(x – 1)(x+1) x+1

9.  3x + 1 x (x – 1)
+
(x – 1)(x+1) x+1 (x – 1)
Now multiply through the denominators
to get the same denominator on both sides
 3x + 1 x2 - x
= +
(x - 1)(x + 1) (x - 1)(x + 1)
Now add the numerators together

10.  x2 + 2x + 1
= Now factor the numerator
(x - 1)(x + 1)
 (x + 1)2
=
(x - 1)(x + 1) And cancel

11. Answer
• x+1
=
x-1

12. Multiplication of Fractions
 When we multiply rational expressions we factor then we cross
cancel.
 x2 - 2x + 1 x2 + 4x + 3
First Factor
x +1 x-1
 (x - 1)2 (x + 3)(x + 1)
= Cancel the x + 1 and the x - 1
x+1 x-1

= (x - 1)(x + 3)

13. Division of Fractions
 Recall that when we divide fractions we multiply by the
reciprocal.
 For example:
 8
9 8 21 4 7 28
= = =
10 9 10 3 5 15
21
 For rational functions we do the same thing. To divide
rational functions, multiply by the reciprocal and then
factor and cross cancel.

14.  x2 - x - 2
x+3 x2 - x - 2 3x+9
=
2x + 2 x+3 2x+2
3x + 9
Times by Reciprocal

15. Answer
 (x - 2)(x + 1) 3(x + 3) 3x-6
= =
x+3 2(x+1) 2
Factor Then cancel

16. Review Test

17. Addition and Subtraction
1.) 3x 4x2
-
2y2 9y
2.) 5 y
+
y y-3
Answer the questions on your own before continuing

18. #1
 (9) 3x 4x2 ( 2y)
-
(9) 2y2 9y (2y)
 =27x 8x²y
-
18y² 18y²

19. #2
5 y
(y-3) + (y)
y y-3
= 5y-15 y²
+
Y²-3y y²-3y
= y²+ 5y – 15
y(y²-3)

20. Multiplication
 1) 2 2 4
5a b b c
3 4 3
b c 25a
 2.) 5 x 7 3x 12
12 3x 7 5 x

21. Answer #1
2 3 4
5a b c 1
3 3 4
25a b c 5a

22. Answer#2
5 x 7 3 x 12
12 3 x 7 5 x
1 1 1

23. Division
1.) x
2
7x 8 6x
2
6x
2 2
x 2x 48 x 7x 6

24. Answer #1
2 2
x 7x 8 x 7x 6
2 2
x 2x 48 6x 6x
(x 8)( x 1) ( x 6)( x 1)
(x 8)( x 6) 6 x( x 1)
( x 8)( x 1) ( x 6)( x 1)
( x 8)( x 6) 6 x( x 1)
x 1
6x

25. Word Problems
 Word problems are easy to solve once you know
what you’re doing.
 Step 1- Read the question and figure out what the
question is asking you to do
 Step 2- label all parts of the question, what you
know, and what you don’t know as “x”

26. Problem 1
Paul can wax his car in 45 minutes. His big
brother John can do the job in 30 minutes. If
they work together, how long will it take them to
wax Paul’s car?

27. Problem 1
x = time to wax the car working
together(minutes)
x x
1
45 30

28. Problem 1
x x
1
45 30
x x
90 1
45 30
x x
90 90 90 1
45 30

29. Problem 1
2x 3x 90
5x 90
x 18 minutes
It will take Paul & John 18 minutes to
wax Paul’s car.

30. Problem 2
It takes one person twice as long to
shovel snow from the driveway as it
takes another using a snow blower. If
the two of them together can clear the
driveway in 8 minutes, how long does
it take the person shoveling alone?

31. Problem 2
x = time for person using the snow
blower to clear the driveway (mins)
2x = time for person shoveling to
complete the driveway (mins)
8 8
1
2x x

32. Problem 2
4 8
1
x x
4 8
x 1
x x

33. Problem 2
4 8 x
x 12 minutes
The person shoveling alone will take 24
minutes the shovel the driveway.

34. If you are having any more trouble with rational
expressions go to the following websites
 http://www.purplemath.com/modules/rtnladd.htm
 http://www.purplemath.com/modules/rtnlmult.htm
 http://faculty.eicc.edu/jmoeller/6_5/sld005.htm
 http://www.mathhelpforum.com/math-help/pre-
calculus/53940-simplifying-rational-expressions-
distance-word-problems.html
 http://www.yourteacher.com/
 http://www.youtube.com/user/yourteachermathhelp
 http://www.ltcconline.net/greenl/courses/152b/Factori
ngRatExpr/FactoringRatExpr.htm