SLC-S22W5//Polynomial and rational expressions

Hello friends, today was another opportunity to participate in the mathematical class on a titled "polynomial and rational expressions", the class was so impactful. So on the below I'm expected to prove my understanding by exercising on the task given.

Without further ado....

Task	1:

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

Answer	...

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Polynomial Expressions

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Polynomial expressions can be defined as the algebraic expressions which consist of variables, coefficients and mathematical terms that can be added, subtracted and multiplied but cannot be divided

[Characteristics]

• Variables: Polynomial expressions contain variables like x, y, or z

• Coefficients: Polynomial expressions have coefficients that are numerical values multiplied by the variables

• Mathematical terms: Polynomial expressions consist of mathematical terms like addition, subtraction and multiplication

• Non divisible: Polynomial expressions cannot be divided by another expression

Let me elaborate with example below...

Examples

• 2X^2 + 3X - 4

• X^3 - 2X^2 + X + 1

• 3X^2 + 2X - 1

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Rational Expressions

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I'm going to define rational expressions as the algebraic expressions which are the ratio of two polynomial expression

[Characteristics]

• Ratio of polynomials: Rational expressions are the ratio of two polynomial expressions

• Variables: Rational expressions contain variables like x, y, or z

• Coefficients: Rational expressions have coefficients which are numerical values multiplied by the variables

[Examples]

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

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Task	2:

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

Answer	...

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Simplifying Rational Expressions

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[Steps to Simplify Rational Expressions]

Factor the numerator and denominator: this method factor both the numerator and denominator to identify common factors

Cancel out common factors: this method Cancel out any common factors between the numerator and denominator

Simplify the expression: this method simplify the resulting expression if possible

[Examples]

Simplify the expression: 1

Here, I'm to simplify the rational expression, (2X^ 2 + 6X )/(X^ 2 + 3X).

Now, I'm factoring the numerator and denominator, (2X (X + 3) ) / (X(X+3 ) ).

Now, cancel out the common factors: 2X / X

Simplify the expression: 2

Simplify the rational expression, ( X^2-4 ) / ( X+2 )

Factor the numerator and denominator, (( X+ 2)(X-2 )) / ( X+2 )

Now, cancel out common factors: x - 2

And then, simplify the expression: x - 2

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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.

Answer	...

[Adding Polynomial Expressions]

Here I'm breaking down into smaller steps...

I am given two polynomial expressions to add them

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

To add those expressions, I followed these steps below.

1. I combined like terms, where I group the terms with the same variable and exponent

2. I Added the coefficients, I add the numerical coefficients of the like terms

[Step by Step Solution]

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

= ( 3X^2+2X^2)+(2X-X)+(1 - 3)

= 5X^2 + X - 2

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[Multiplying Polynomial Expressions]

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I am given two polynomial expressions

(2X+3) and (X-2)

To multiply these expressions I followef these steps below to arrive at my result

1. I used the distributive property, where I multiply each term in the first expression by each term in the second expression

2. I combine like terms, I grouped the terms with the same variable and exponent

[Step by Step Solution]

(2X+3) × (X-2)

= 2X(X-2) + 3(X-2)

= 2X^2-4X + 3X-6

= 2x^2-x - 6

My final answer is ➡️ 2X^2- X - 6

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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 your final outcome)

Answe	1:

Let's write a polynomial expression for the total cost(c) of buying x packets of beads with a 10% discount

c = (5+2x) - 0.1(5+2x)

= 0.9(5+2x)

= 4.5+1.8x

Now, calculate the total cost if Ali buys 5 packets of beads

c = 4.5 + 1.8(5)

= 4.5 + 9

= 13.5 ✅

[Steps]

Step 1: Write the initial cost expression

Initial cost = fixed cost+variable cost

= $5+$2x

Step 2: Apply the 10% discount

Discount = 10% of Initial cost

= 0.1($5+$2x)

Step 3: Subtract the discount from the initial cost

Total cost (c) = Initial cost-Discount

= ($5+$2x) - 0.1($5+$2x)

= 0.9($5 + $2x)

= 4.5+1.8x

Step 4: Calculate the total cost for 5 packets of beads

C = 4.5+1.8(5)

= 4.5+9

= 13.5 ✅

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.

I'm writing a rational expression for the ratio of wheat to total harvest

Wheat/Total Harvest = x / (x + 3x)

= x/4x

= 1/4

Now, calculated the ratio of wheat to total harvest if the farmer harvests 4 tons of wheat

Ratio = 1/4

= 0.25✅

[Steps]

Step 1 Write the ratio expression

Wheat / to Total Harvest = x / (x+3x)

Step 2 Simplify the denominator
x+3x = 4x

Step 3: Rewrite the ratio expression with the simplified denominator
Wheat / Total Harvest = x / 4x

Step 4: Cancel out the common factor (x)

Wheat/Total Harvest= 1/4

Step 5: Calculate the ratio of wheat to total harvest for 4 tons of wheat

Ratio= 1 / 4

= 0.25 ✅

Thank you all for your time I believe we've learned together, see you guys in our next class.

Best regards to the Teacher: @khursheedanwar