![Add a heading.jpg](https://cdn.steemitimages.com/DQmeefA4PwoBWB2DjaWpbVFEQswT7vXsoTkKApPfGKKcJEf/Add%20a%20heading.jpg) _{Photo taken from [Pixabay](https://pixabay.com/photos/math-work-mathematics-formulas-4711302/)} Hello everyone. Today I will take part in week 5 of Steem Learning Challenge, by @khursheedanwar, based on polynomial and rational expressions. You can check it [Steemit learning challenge S22W5//Polynomial and rational expressions](https://steemit.com/algebra-s22w5/@khursheedanwar/steemit-learning-challenge-s22w5-polynomial-and-rational-expressions). So, let's get started. ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) ![task1.jpeg](https://cdn.steemitimages.com/DQmYNVWTMh12GHsRuvZrPiEZbJ4Cok6G2acmQ8wKf3aQisz/task1.jpeg) |Explain difference between polynomial and rational expressions .Provide examples of each type of system of equation and describe their general forms.| |--| Before we can highlight the differences between polynomial and rational expressions, we need to give an overview of them.

Polynomial Expressions

A **polynomial expression** can be defined as an algebraic expression that contains one or more terms which contain: - a variable that is raised to a positive integer power - a coefficient A polynomial is continuous across the entire domain.

General form of a polynomial expression

P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ where: - a_n, a_n-1, ... a₀ are coefficients - n is a positive integer which is also known as the degree of the polynomial Examples: - **3x² + 2x + 1** is a 2nd degree polynomial expression - **x⁴ - 7x²+ 1x - 2** is a 4th degree polynomial expression

Rational Expressions

A **rational expression** can be defined as a fraction which contains polynomials in both the numerator and denominator. A key information is that **the denominator cannot be zero**, as a division by zero is undefined.

General form of a rational expression

R(x) = P(x)/Q(x) where: - P(x) and Q(x) are polynomials - Q(x) ≠ 0 Examples: - **(x² + 3x)/(x - 1)** - **(3x² - 5)/(x² + 2x - 3)**

Real world applications

The polynomial expression is used in: - physics: can predict the maximum height of a thrown ball, like the example we had in Week 4, - economics: to determine the amount of items sold for maximum profit - geometry: to calculate areas The rational expression is used in: - medicine: to determine the amount of medication dosage for a certain body weight - engineering: can help determine the total resistance of parallel circuits

Differences between Polynomial Expressions and Rational Expressions

Some of the key differences between the 2 types of expressions can be seen in the table below: | Feature | Polynomial Expressions | Rational Expressions | |------------|:----------------------------------------------------------------------------:|:-------------------------------------------------------------------:| | Definition | An expression that contains terms with variables raised to a positive number | A fraction that has 2 polynomials: in the numerator and denominator | | Degree | Highest power of the variable | Degree depends on both the numerator and denominator | | Form | P(x) = anx^n + an-1x^n-1 + ... + a1x + a0 | R(x) = P(x)/Q(x) | | Domain | The domain is all real numbers R | Domain excludes values where Q(x) = 0 | | Continuity | Continuos for all real numbers | Contains discontinuities when Q(x) = 0 | | Graph | A smooth curve | A curve that may have vertical asymptotes or holes | | Examples | 3x^2 + 4x + 5; x^3 - 2x + 1 | (x + 2)/(x - 1); (x^2 - 1)/(x^2 + x - 6) | ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) ![task2.jpeg](https://cdn.steemitimages.com/DQmWY31bznznvSki36CFxFAkmq6vwdo8p6fG4Y1FCTpaXHB/task2.jpeg) |Explain steps used in simplifying a rational expression. Write some common factors required to be cancel out?| |--| For this task, let's consider the following rational expression: ``` (x^2 - 9)/(x^2 - 6x + 9) ```

Step 1:

The first step in simplifying this expression is to factorize the numerator and denominator: ``` The numerator x^2 - 9 can be identified as a difference of squares: x^2 - 9 = (x - 3)(x + 3) The denominator is a perfect square: x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2 ``` After this step, we can rewrite the expression as: ``` (x^2 - 9)/(x^2 - 6x + 9) = (x - 3)(x + 3)/(x - 3)^2 ```

Step 2:

Next step is to cancel common factors. We can identify the common factors in our expression and cancel them out by dividing the numerator and denominator by the common factor. In our example, both the numerator and denominator contain ```(x - 3)``` as a factor. We can cancel ```(x - 3)``` once from both: ```(x - 3)(x + 3)/(x - 3)^2 = (x + 3)/(x - 3), where x ≠ 3```

Step 3:

An important step is to state the restrictions so we avoid having the denominator equal 0. In this example, the original denominator was ```(x - 3)^2``` which means that ```x - 3 = 0``` **is NOT allowed**. ``` Restriction: x ≠ 3 ```

Step 4:

We can now write the final simplified expression: ```(x^2 - 9)/(x^2 - 6x + 9) = (x + 3)/(x - 3); where x ≠ 3``` The most common factors that can be canceled out are: - Binomials (x - a), (x - b), ... Example: ```(x + 3)(x - 2)/(x - 2)(x +4) → (x + 3)/(x + 4); where x ≠ 2, 4``` - Monomials x, 2x, 3x^2, ... Example: ```4x^3/2x^2 → 2x/1; where x ≠ 0 ``` ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) ![task3.jpeg](https://cdn.steemitimages.com/DQmP8RCbyahLNd3DEh33JuqExd4gk9sMuruWC3ay1xgVYDS/task3.jpeg) |Please add these polynomial expressions 3x^2 + 2x + 1 and 2x^2 - x - 3 and share your final expression.| |--| To perform the addition of the given polynomials: 3x^2 + 2x + 1 and 2x^2 - x - 3 we need to follow these steps: ``` Write them side by side: (3x^2 + 2x + 1) + (2x^2 - x - 3) By getting rid of the parenthesis we get: 3x^2 + 2x + 1 + 2x^2 - x - 3 We can group them like this: ( 3x^2 + 2x^2) + (2x - x) + (1 - 3) Combining x^2 terms: 3x^2 + 2x^2 = 5x^2 Combining x terms: 2x - x = x Combining constants: 1 - 3 = -2 ``` Now, we can write the result: ```5x^2 + x - 2``` Final answer: **5x^2 + x - 2** The solve can also be seen in the image below: ![image.png](https://cdn.steemitimages.com/DQmRQ636KDhpZjuYVLwuhJYvoZoYTwY2kNoZ1hWC5zj5vHE/image.png) ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) |Share multiplication of polynomial expressions 2x + 3 and x - 2 with final outcome of resulting expression.| |--| To perform the multiplication of the given polynomials: 2x + 3 and x - 2 we need to follow these steps: - Step 1: Use the distributive property to multiply each term in 2x + 3 by each term of x - 2: ``` (2x + 3) * (x - 2) = 2x(x - 2) + 3(x - 2) ``` - Step 2: Perform the multiplications: ``` 2x(x - 2): multiply 2x * x = 2x^2 and 2x * (-2) = -4x 3(x - 2): multiply 3 * x = 3x and 3 * (-2) = -6 ``` By combining these terms, we have: ```2x^2 - 4x + 3x - 6``` Combining the like terms: ``` 2x^2 remains 2x^2 -4x + 3x = -x -6 remains -6 ``` The final result can be written: ```2x^2 - x -6```. The solve can also be seen in the image below: ![image.png](https://cdn.steemitimages.com/DQmYxqGXinwkUv5THQcDCQe6z4t6XnGoMDJWUeFtTNd3VMG/image.png) ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) ![task4.jpeg](https://cdn.steemitimages.com/DQmedc1XH81DnrYPsQcHfNuVEwwUba1H4eLY6DmPksK91qs/task4.jpeg)

Scenario 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.| |--| Before starting to solve this, **we need to identify the given information that will help us**: ``` Fixed cost: 5 Cost per packet: 2x; where x is the number of packets Discount: 10% or 0.1 ``` With this written down, we can write the **total cost expression before the discount**: ```Cost before discount = 5 + 2x; where x is the number of packets``` If we apply the discount to the cost, **the expression becomes**: ``` Total Cost (C) = 1(5 + 2x) - 0.1(5 + 2x) C = 0.9(5 + 2x) ``` Now that we have the total cost (C) expression for buying x packets with a discount of 10%, we **calculate the total cost for 5 packets**: ``` C = 0.9(5 + 2(5)) C = 0.5(5 + 10) C = 0.9(15) C = 13.5 ``` **Final answer:** The total cost of buying 5 packets with a fixed price of $5 and a cost of $2 per packet is **$13.50**. **Conclusion: Ali has to spend **$13.50** to buy 5 packets of beads.** The solve can also be seen in the image below: ![image.png](https://cdn.steemitimages.com/DQmW6ir5qg2sZfKNvSumgGFPfKDhGB153ruWNk9KS2v4fHC/image.png) ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png)

Scenario 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.| |--| As we did for Scenario 1, we need to write the information that is given to us: ``` Amount of wheat harvested: x Amount of barley: 3x ``` We can now write the expression for the total amount harvested: ``` Total harvest = x + 3x Total harvest = 4x ``` The ratio ratio of wheat to total harvest can be written like: ``` Ratio = x/4x Which gets simplified to: Ratio = 1/4, for x different than 0 ``` If the farmer is harvesting 4 tons of wheat, the ratio will still be 1/4 but let's also check in the ratio expression: ``` Ratio = 4/(4*4) Ratio = 4/16 Ratio = 1/4 Or: Ratio = 0.25 ``` **Conclusion: The ratio of wheat to total harvest will always be 1/4 or 0.25 as long as x is different than 0**. The solve can also be seen in the image below: ![image.png](https://cdn.steemitimages.com/DQmWCfPDWEn96gGkehq1LbH3sHH6556CngE3Vd63x7oAijb/image.png) ![spacer.png](https://cdn.steemitimages.com/DQmYJLRxbELh1CCtUjRFMH4DS1NoQGDm1XTP5FcxVukouFw/spacer.png) As always, thank you for reading and I'd like to extend an invitation to @ady-was-here, @radudangratian and @cmalescov to take part in this.