rational expressions
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PRE-CALMR. WATT
BY D A N N Y C O L L I C U TT, R U S T I N E PA D I L L A , J U N E B E C O R O N I A
Rational Expressions and Equations
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
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)
Example
Now try : 3x2 - 4x/ 2x2 - x
Click to see the answer and the steps used
Answer
3x2 - 4x x(3x - 4) 3x - 4 = = 2x2 - x x(2x - 1) 2x - 1
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.
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.
Example
3x + 1 x + x2 - 1 x + 1
3x + 1 x + (x – 1)(x+1) x + 1
First Factor everything and find the lowest common denominator
The LCD is (x - 1)(x + 1)
3x + 1 x (x – 1) + (x – 1)(x+1) x + 1 (x – 1)
3x + 1 x2 - x = + (x - 1)(x + 1) (x - 1)(x + 1)
Now multiply through the denominatorsto get the same denominator on both sides
Now add the numerators together
x2 + 2x + 1 = (x - 1)(x + 1)
(x + 1)2 = (x - 1)(x + 1)
Now factor the numerator
And cancel
Answer
• x + 1 = x - 1
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)
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.
x2 - x - 2 x + 3 x2 - x - 2 3x+9 = 2x + 2 x + 3 2x+2 3x + 9 Times by Reciprocal
Answer
(x - 2)(x + 1) 3(x + 3) 3x-6 = = x + 3 2(x+1) 2
FactorThen cancel
Review Test
Addition and Subtraction
1.) 3x 4x2 - 2y2 9y
2.) 5 y + y y - 3
Answer the questions on your own before continuing
#1
(9) 3x 4x2 ( 2y) - (9) 2y2 9y (2y)
=27x 8x²y - 18y² 18y²
#2
5 y (y-3) + (y) y y - 3
= 5y-15 y² + Y²-3y y²-3y
= y²+ 5y – 15 y(y²-3)
Multiplication
1)
2.)
3
42
43
2
25
5
a
cb
cb
ba
x
x
x
x
57
123
312
75
Answer #1
433
432
25
5
cba
cba
a5
1
Answer#2
x
x
x
x
57
123
312
75
11 1
Division
1.)
67
66
482
872
2
2
2
xx
xx
xx
xx
Answer #1
xx
xx
xx
xx
66
67
482
872
2
2
2
)1(6
)1)(6(
)6)(8(
)1)(8(
xx
xx
xx
xx
)1(6
)1)(6(
)6)(8(
)1)(8(
xx
xx
xx
xx
x
x
6
1
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”
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?
Problem 1
x = time to wax the car working together(minutes)
13045
xx
Problem 1
13045
xx
1
304590
xx
19030
9045
90 xx
Problem 1
9032 xx
905 x
utesx min18
It will take Paul & John 18 minutes to wax Paul’s car.
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?
Problem 2x = time for person using the snow blower to clear the driveway
(mins)2x = time for person shoveling to complete the driveway (mins)
18
2
8
xx
Problem 2
184
xx
1
84
xxx
Problem 2
x 84
utesx min12
The person shoveling alone will take 24 minutes the shovel the driveway.
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/FactoringRatExpr/FactoringRatExpr.htm
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