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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to multiply and divide rational expressions.</para>

Overview

  • Multiplication Of Rational Expressions
  • Division Of Rational Expressions

Multiplication of rational expressions

Rational expressions are multiplied together in much the same way that arithmetic fractions are multiplied together. To multiply rational numbers, we do the following:

Method for Multiplying Rational Numbers
  1. Reduce each fraction to lowest terms.
  2. Multiply the numerators together.
  3. Multiply the denominators together.

Rational expressions are multiplied together using exactly the same three steps. Since rational expressions tend to be longer than arithmetic fractions, we can simplify the multiplication process by adding one more step.

Method for Multiplying Rational Expressions
  1. Factor all numerators and denominators.
  2. Reduce to lowest terms first by dividing out all common factors. (It is perfectly legitimate to cancel the numerator of one fraction with the denominator of another.)
  3. Multiply numerators together.
  4. Multiply denominators. It is often convenient, but not necessary, to leave denominators in factored form.

Sample set a

Perform the following multiplications.

3 4 · 1 2 = 3 · 1 4 · 2 = 3 8

8 9 · 1 6 = 8 4 9 · 1 6 3 = 4 · 1 9 · 3 = 4 27

3 x 5 y · 7 12 y = 3 1 x 5 y · 7 12 4 y = x · 7 5 y · 4 y = 7 x 20 y 2

x + 4 x - 2 · x + 7 x + 4 Divide out the common factor  x + 4. x + 4 x - 2 · x + 7 x + 4 Multiply numerators and denominators together . x + 7 x - 2

x 2 + x - 6 x 2 - 4 x + 3 · x 2 - 2 x - 3 x 2 + 4 x - 12 . Factor . ( x + 3 ) ( x - 2 ) ( x - 3 ) ( x - 1 ) · ( x - 3 ) ( x + 1 ) ( x + 6 ) ( x - 2 ) Divide out the common factors  x - 2  and  x - 3. ( x + 3 ) ( x - 2 ) ( x - 3 ) ( x - 1 ) · ( x - 3 ) ( x + 1 ) ( x + 6 ) ( x - 2 ) Multiply . ( x + 3 ) ( x + 1 ) ( x - 1 ) ( x + 6 ) or x 2 + 4 x + 3 ( x - 1 ) ( x + 6 ) or x 2 + 4 x + 3 x 2 + 5 x - 6 Each of these three forms is an acceptable form of the same answer .

2 x + 6 8 x - 16 · x 2 - 4 x 2 - x - 12 . Factor . 2 ( x + 3 ) 8 ( x - 2 ) · ( x + 2 ) ( x - 2 ) ( x - 4 ) ( x + 3 ) Divide out the common factors 2,  x + 3  and  x - 2. 2 1 ( x + 3 ) 8 4 ( x - 2 ) · ( x + 2 ) ( x - 2 ) ( x + 3 ) ( x - 4 ) Multiply . x + 2 4 ( x - 4 ) or x + 2 4 x - 16 Both these forms are acceptable forms of the same answer .

3 x 2 · x + 7 x - 5 . Rewrite  3 x 2  as  3 x 2 1 . 3 x 2 1 · x + 7 x - 5 Multiply . 3 x 2 ( x + 7 ) x - 5

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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