Hello friends, I hope everyone is well? Alhamdulillah I am fine.
I am @sheikhtuhin
From #Bangladesh

Dear friends, today I am going to join Steemit learning challenge S22W5//Polynomial and rational expressions.This learning challenge is organized by the beloved teacher of this platform, @khursheedanwar Sir.I am very grateful to Sir because now I don't need to practice Maths. I have forgotten it for a long time, but thanks to Sir's arrangement, I am able to try those old topics again.

Without further ado, I'll start today's homework right away.

Polynomials and rational expressions are two important concepts in mathematics. The difference between them lies mainly in their structure and usage.

The steps used to simplify a logical expression are explained and some common reasons for cancellation are listed below.

Steps to simplify logical expressions:

(i) Analysis of numerator and denominator.
(ii) Abolish general positions.
(iii)Combine with LCM if necessary.
(iv) Identify voidable conditions
(v) Simplified expression.

Analysis of numerator and denominator.

Dividing the numerator and denominator into their common multiples reduces the complexity of polynomial fractions.

For example:

            X²-4/x²-x-6
         = X²-2² / x²-3x+2x-6  [The numerator is the formula for a²-b² and the denominator is the middle factor.]
         = (x+2)(x-2) / x(x-3)+2(x-3)
         = (X-2)(X-2) / (X-3)(X+2)

Abolish general positions.

If there are common factors between the numerator and denominator, they must be canceled.

For example:

               (x-2)(x+1) / (X-4)(x+1)
            = (x-2) / (X-4) [where x ≠ -1]

Combine with LCM if necessary.

If there are two or more rational numbers, they must be converted into a single fraction by taking the lowest multiple of their denominators.

For example:

                   1/(x+1)  +  2/( x-1)
                 = 1( x-1)+ 2(x+1)/ (x+1)(x-1)[Adding fractions]
                 = X-1+2x+2 / x²-1² [By applying the formula of a²-b² in the denominator]
                = 3x+1/ x²-1

Identify voidable conditions

If any multiple of the numerator or denominator is zero, the expression becomes undefined, excluding such values from the domain.

For example:

                      3x+1/ (x+1)(x-1)

    If x+1=0 and x-1=0 then x≠1,-1

Simplified expression.

The final expression obtained after cancellation and simplification must be written down and this will be considered final.

Solution:

3x²+2x+1 and 2x²-x-3
These two expressions need to be added.

First step:- The paths of each polynomial need to be sorted, but the above two expressions are sorted.

              3x²+2x+1
              2x² - x - 3

Step-2: Similar paths need to be added.

    3x²+2x²
  = 5x²

  2x-x 
= x

  1-3
= -2

Step-3: After adding the similar terms, we add all the results together to get

        5x² + x - 3

Final result: 5x² +x -3

                                    (Ans:)

The polynomials 2x + 3 and x - 2 are multiplied and divided by the final result.

Multiplication :

         ( 2x+3)×(x-2)

Step-1: Each term must be multiplied with each other.

              2x.x
            = 2x²

              2x.(-2)
           = -4x

             3.x
          = 3X


             3.(-2)
          = -6

Step-2: The products must be added.

                2x²+(-4x)+3x+(-6)
             = 2x²-4x+3x-6

step-3 : One type of word must be combined.

              2x²-4x+3x-6
            = 2x²-x-6

Final result:

               2x²-x-6
                           (Ans.)

Quotient:

   Let's divide,
                    ( 2x+3)/ (x-2)

The numerator (2x+3) and the denominator (x−2) have no common multiples. So it does not simplify.

Final result:

                         ( 2x+3)/ (x-2)

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.

Solution:

I will try to solve it step by step according to the information given above, God willing.

Step-1: Find out the cost of each packet.

Price per packet

                               = 5$ +2$ = 7$

So the total cost for "X" packets is,
C= 7x

Step-2: Considering 10% discount

• A 10% discount means you have to spend 90% of the total cost.

                 So,
                      For discounted prices,
                                          Cd= C× 90%
                                        => Cd = C × 90/100
                                        => Cd = C×0.90
  

Step-3 : C= 7x Set the standard

                          Cd= 7x×0.90
                            Cd=  6.3x

step-4 : Figuring out the cost for 5 packets of beads

5 packets means the value of X is 5.

                          Cd = 6.3x
                        =>Cd= 6.3×5
                            Cd= 31.5       [Here Cd means Cdiscount.]

Final result=

• Total cost expression, C= 7x

• Expression of cost with 10% discount, Cd=6.3x

• Total cost for five packets, Cd= 31.5

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.

Solution

Step-1: Determine the total harvest

The farmer is harvesting ,
x tons of wheat and 3x tons of barley.

Total crop,
= X+3x
= 4x

Step 2: Ratio of wheat to total crop.

                  Quantity of wheat = X
                   Total harvest = 4X

Wheat ratio,
= Wheat quantity/total crop.
X/4X

Step 3: Simplification.

Common terms of numerator and denominator cancel (if x≠0)

            x/4x = 1/4

Therefore the ratio of wheat = 1:4

Step 4:

Find the ratio for (x=4) tons of wheat:

If the farmer harvests 4 tons of wheat, the total harvest,
4x
= 4×4
= 16 tons

Wheat ratio:
4:16
= 4/16
= 1/4 or 0.25

Final answer:

Therefore, the ratio of wheat to the farmer's total harvest is always 1/4 or 0.25, which is a simplified form.