SLC-S22W5//Polynomial and rational expressions

Hello, friends
I so elated to join this great learning section, here is my own contribution.

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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.

Polynomial Expressions

Polynomial expression is an algebraic expression consisting of variables and constants, involving addition, subtraction and multiplication, but division is not applicable.

• General Form
  an×x^n + an – 1x^n-1 + ........ + an1x + a0

Where;
n = non-negative integer
an – 1, ..... a0 = coefficients

• Examples:
  (1). 3x² + 2x + 1

(2). 4x³ – x² + 5x – 7

Rational Expressions

Rational expression is an expression which is the ratio of two polynomial expressions.

• General Form
  P(x) / Q(x)

Where;
P(x) and Q(x) are polynomials
and Q(x) ≠ 0

• Examples:
  (1). (x² + 3x + 2) / (x + 1)

(2). (4x² + 5) / (X² – 9)

Differences

Polynomial Expressions	Rational Expressions
They are single expressions	They are ratios of two expressions
They do not have variables in the denominator	They have variables in the denominator
Radical expressions are not present	Radical expressions may not be present
Variables are raised to the power of the non-negative integer	The denominator is not equal to zero

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Task 2

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

Steps used in simplifying a rational expression

• Firstly, factor both the numerator and the denominator.

• Cancel the common factors on both the numerator and denominator.

• After cancelling rewrite the expression with the remaining factors.

• Lastly, solve to get the simplified expression.

Example
Simplify: (x² – 4) / (x² + 3x + 2)
factor both the numerator and denominator

• x² – 4 = (x – 2)(x + 2)
• x² + 3x + 2 = (x + 1)(x + 2)
  cancel the common factors
  (x – 2)(x + 2) / (x + 1)(x + 2)
  = (x – 2) / (x + 1)

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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.

(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)

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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 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.

Scenario 1

Ali's craft store

Given;
Fixed cost = $5
Cost per packet = $2
Total cost before discount; C(x) = 5 + 2x
Cost after 10% discount; C(x) = (5 + 2x) × 0.9

C(x) = 0.9(5 + 2x) = 4.5 + 1.8x

• C(x) = 4.5 + 1.8x

solving for 5 packets
Substitute; x = 5 into the above expression

C(5) = 4.5 + 1.8(5)
= 4.5 + 9 = 13.5
Total cost (C) = $13.50

Scenario 2

Farmer's harvest

Given;
Wheat = x tons
Barley = 3x tons
Total harvest = x + 3x = 4x
Ratio of wheat to the total harvest = x / 4x

cancel 'x' in the numerator and denominator
Ratio = x / 4x = 1 / 4

Solving for 4 tons of wheat
substitute x = 4 into the ratio

Ratio = x / 4x = 4 / 4(4) = 4 / 16 = 1 / 4

Ratio of wheat to total harvest = 1 / 4

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I will invite;
@imohmitch
@us-andrew
@precious9

Cc,
@khursheedanwar