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- Summary of key concepts
<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>This module presents a summary of the key concepts of the chapter "Rational Expressions".</para>
Summary of key concepts
Rational expression (
[link] )
A
rational expression is an algebraic expression that can be written as the quotient of two polynomials. An example of a rational expression is
Domain of a rational expression (
[link] )
The
domain of a rational expression is the collection of values for which the raticlnal expression is defined. These values can be found by determining the values that will not produce zero in the denominator of the expression.
The domain of
is the collection of all numbers except
.
Equality property of fraction (
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- If
, then
.
- If
, then
.
Negative property of fractions (
[link] )
Reducing a rational expression (
[link] )
- Factor the numerator and denominator completely.
- Divide the numerator and denominator by any factors they have in common.
Common cancelling error (
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Since
is not a common factor, it cannot be cancelled.
Multiplying rational expressions (
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- Factor all numerators and denominators.
- Reduce to lowest terms first by dividing out all common factors.
- Multiply numerators together.
- Multiply denominators together.
It will be more convenient to leave the denominator in factored form.
Division of rational expressions (
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Building rational expressions (
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Building rational expressions is exactly the opposite of reducing rational expressions. It is often useful in adding or subtracting rational expressions.
The building factor may be determined by dividing the original denominator into the new denominator. The quotient will be the building factor. It is this factor that will multiply the original numerator.
Least common denominator lcd (
[link] )
The LCD is the polynomial of least degree divisible by each denominator. It is found as follows:
- Factor each denominator. Use exponents for repeated factors.
- Write each
different factor that appears. If a factor appears more than once, use only the factor with the highest exponent.
- The LCD is the product of the factors written in step 2.
Fundamental rule for adding or subtracting rational expressions (
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To add or subtract rational expressions conveniently, they should have the same denominator.
Adding and subtracting rational expressions (
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Note that we combine
only the numerators.
Rational equation (
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A
rational equation is a statement that two rational expressions are equal.
Clearing an equation of fractions (
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To clear an equation of fractions, multiply both sides of the equation by the LCD. This amounts to multiplying every term by the LCD.
Solving a rational equation (
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- Determine all values that must be excluded as solutions by finding the values that produce zero in the denominator.
- Clear the equation of fractions by multiplying every term by the LCD.
- Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.
- Check the solution by substitution.
A potential solution that has been excluded because it creates an undefined expression (perhaps, division by zero) is called an
extraneous solution.
Questions & Answers
differentiate between demand and supply
giving examples
differentiated between demand and supply using examples
Lambiv
how is the graph works?I don't fully understand
hi guys good evening to all
Lambiv
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
Can I ask you other question?
Shukri
What is different between quantity demand and demand?
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
how do you save a country economic situation when it's falling apart
what is the difference between economic growth and development
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
any question about economics?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
What is the difference between perfect competition and monopolistic competition?
Mohammed
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Source:
OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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