Steemit learning challenge S22W5//Polynomial and rational expressions

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Greetings

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I warmly welcome every steemian in week 5 course about algebra!
It's important to discuss introduction of algebra and important terminologies about algebra before getting involved into in depth of main topic that I am teaching you guys today!

Algebra is basically a major branch of mathematics which consists of study of variables and their relationships with eachother.If I talk about algebra then It includes usage of different symbols, equations, and formulas for solving problems and for modeling real world applications.

Defining polynomial and rational expressions

Polynomial expressions

If I talk about polynomial expression then this is an algebraic expression which contains variables, coefficients as well as mathematical terms,Basically it contains those terms which may be added, subtracted and multiplied but they can't be divided.

Practical examples

• Expenses for fruits are equal to 2 apples + 3 bananas
• Area covered by a room is equal to 2(length + width)

Algebraic examples

• 2x^2 + 3x - 4
• x^3 - 2x^2 + x + 1

Rational expression

If I talk about rational expression then it's an algebraic expression which is ratio of two polynomial expressions together.

Practical examples

• Fuel efficiency can be measured by dividing distance with fuel used so it can be written as distance traveled ÷ amount of fuel used
• Speed can measured by dividing distance with time so it may be written as distance ÷ time

Algebraic examples

• (2x^2 + 1) / (x - 3)
• (x^2 - 4) / (x + 2)

Adding, subtracting and multiplying polynomial expressions

Adding polynomial expressions

If you have have to add polynomial expressions then you should follow the following steps so;

Combine like terms together!

Example 1

(2x^2 + 3x - 4) + (x^2 - 2x + 1)

= (2x^2 + x^2) + (3x - 2x) + (-4 + 1)

= 3x^2 + x - 3

Example 2

(x^3 - 2x^2 + x + 1) + (2x^3 + x^2 - 3x - 2)

(x^3 + 2x^3) + (-2x^2 + x^2) + (x - 3x) + (1 - 2)

3x^3 - x^2 - 2x - 1

Subtracting polynomial expressions

If you need for subtracting polynomial expression then you have to follow following steps;

Distribution of negative sign first and then combining similar terms together!

Example

(2x^2 + 3x - 4) - (x^2 - 2x + 1)

(2x^2 - x^2) + (3x + 2x) + (-4 - 1)

x^2 + 5x - 5

Multiplying polynomial expressions

If you have to multiply polynomial expressions then you should follow the following examples;

You need for using distributive property here!

Example

(2x^2 + 3x - 4) × (x + 2)

2x^2(x + 2) + 3x(x + 2) - 4(x + 2)

2x^3 + 4x^2 + 3x^2 + 6x - 4x - 8

2x^3 + 7x^2 + 2x - 8

Simplifying rational expressions by factoring and cancelling

Simplification steps

Factoring of numerator and denominator

If I talk about first step then here you need for factoring numerator and denominator both from rational expression in their prime factors.

Cancellation of common factors

In second step you need for cancelling out any factor which is common in between numerator and denominator.

Writing of simplified expression

Last but not least,it's a step for writing simplified expression.

Step by step rational expressions simplification by using methods of factoring and canceling

Simplification of a simple fraction(6x^2) / (3x)

• First step is to factor numerator and denominator so after factoring of numerator and denominator in their prime factors i got;

(2×3×x×x) / (3×x)

• According to my description of steps now we have for cancelling out any common factor present in between numerator and denominator so it can be written as;

(2×x) / 1

• If I talk about third step then it's all about writing simplified expression so simplified expression may be written as;

2x

Simplification of fraction with quadratic expression (x^2 - 4) / (x + 2)

• First step is to factor numerator and denominator so after factoring of numerator and denominator in their prime factors i got;

((x + 2)(x - 2)) / (x + 2)

• According to my description of steps now we have for cancelling out any common factor present in between numerator and denominator so it can be written as;

(x - 2) / 1

• If I talk about third step then it's all about writing simplified expression so simplified expression may be written as;

x - 2

Applying polynomial and rational expressions in real world problems

Applying Polynomial and Rational Expressions in Real-World Problems:

Polynomial expression real world applications

Cost calculation

Suppose if there's a company producing boxes of cereal with expenses function then it can be represented in form of this polynomial expression which is 2x^2 + 5x + 10 and here x are number of boxes which are producing.

Optimization

If there's a farmer having 100 meters of fencing for enclosing a rectangular field so area for field may be represented by polynomial expression which is A = x(100 - 2x)and here x is length of field.

Rational expressions real world applications

Physics and engineering

Suppose if speed or velocity of an object then rational expressions can be helpful for representing it as v = (2x^2 + 3x - 1) / (x + 1) and here x is time.

Economics

Suppose if there's demand for particular product and you have to represent it in rational expression then you can represent it as D=(2x^2 - 5x + 1) / (x - 1) and here D is showing demand and x is representing price.

By implementing polynomial and rational expressions we may model and analyze real-world problems so that outcomes may be optimized and decision may be made accordingly!

Step by step solution of equations containing polynomial and rational expressions

If you have to solve equations which include polynomial and rational expressions then factoring , cancelling as well as other algebric techniques would be used!

Practical explanation with steps

Solving a polynomial equation which is x^2 + 4x - 5 = 0

• First there's a need of factoring a polynomial as x^2 + 4x - 5 = (x + 5)(x - 1) = 0
• Secondly you should set each factor equivalent to zero as I am doing; x + 5 = 0 or x - 1 = 0
• Last but not least you need to solve for x as; x = -5 or x = 1

Solving a rational equation(2x + 1) / (x - 2) = 3

• First you have for applying cross-multiplication technique after which I got; 2x + 1 = 3(x - 2)
• Second step is all about distribution and simplification so it will be written as 2x + 1 = 3x - 6
• Last but not the least there's a need for isolating for x so it will be written as x = 7

I must say that if you will follow these steps and understand these algebric techniques and methods then you will solve those equations easily in which there are polynomial and rational expressions!

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Homework tasks

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•	Task 1

• Explain difference between polynomial and rational expressions .Provide examples of each type of system of equation and describe their general forms.

•	Task 2

• • Explain steps used in simplifying a rational expression.Write some common factors required to be cancel out?

•	Task 3

• Please add these polynomial expressions 3x^2 + 2x + 1 and 2x^2 - x - 3 and share your final expression.

• Share multiplication of polynomial expressions 2x + 3 and x - 2 with final outcome of resulting expression.

(You are required to solve these problems at paper and then share clear photographs for adding a touch of your creativity and personal effort which should be marked with your username)

•	Task 4

Scenario number 1

Suppose if there's a person named Ali have craft store and he is selling beads in x packet which have fixed cost of $5 plus $2 for each packet.Now you have to write polynomial expression for representing total cost(C)of buying for x packets of beads by considering that there's a 10% discount+Also you have for calculating total cost of Ali buys 5 packets of beads.

(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)

Scenario number 2

Suppose there's a farmer harvesting x tons of wheat and 3x tons of barley.Now you need to write a rational expression for representing ratio of wheat to total harvest in which there's wheat and barley and you have to simplify expression also at end+You also need for calculating ratio of wheat to total harvest if farmer is harvesting 4 tons of wheat.

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Rules to participate

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Follow the below mentioned rules for participating in this learning challenge!

• You can post from your own blog only.

• Your content must be original and must be steemexclusive.

• Your post should contain minimum of 500 words.

• AI generated content or plagiarized content is strictly prohibited!

• You can participate irrespective of your #clubstatus

• You must use your own images or only created with Canva.

• Mention my username @khursheedanwar at least one time in your post!

• Your title should be - "SLC-S22W5//Polynomial and rational expressions."

Tags should be - #algebra-s22w5 , #learningchallange , #clubstatus , #countryname , #steemexclusive , #learnwithsteem etc

Contest deadline is from 13th Janauary to 19th Janauary.

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Evaluation criteria

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Criteria	Status
Plagiarism 🆓	✅
AI 🆓	✅

Category	Remark
Task 1	/1.5
Task 2	/1.5
Task 3	/3
Task 4	/3
Post quality	1
Total grade	/10

Teacher's Observation

Task 1 review	-----
Task 2 review	-----
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Recommendations or suggestions for improvement of content

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Winners selection and rewards distribution

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Winners will be selected solely at ratings that they receive according to efforts they put in their post.All posts will be evaluated before 20 hours after posting and ratings will be clearly defined according to performance of each task by participants!At end of week,I will make sure to select top 4 users with their excellent performance in respective week.Number of comments will not be any consideration for choosing winners!

Steemcurator01 and steemcurator02 will distribute their voting power effectively throughout the week but their votes at each participant's post will not be guaranteed!
Top 4 selected users will be eligible to receive extra prize vote from steemcurator01.

Cc;

@disconnect
@pennsif
@steemchiller

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Thanks

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