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Explain the difference between polynomial and rational expressions. Provide examples of each type of system of equation and describe their general forms
Polynomial Expression
A polynomial expression refers to an algebraic mathematical expression that is made up of variables, arithmetic operations (+, -, ×) as well as coefficients in which the index or power of the variables are not negative integers.
This implies that polynomial expressions are characterized by variables which could be any letter of the alphabet, coefficients which are constants by which the variables are multiplied, powers or indices which are positive integers indicating the exponent the variable is raised to, and arithmetic operations which are the basic operations in mathematics including addition, multiplication, and subtraction.
Polynomial expression takes the general form of mx² + ny - o
where x and y are variables, m and n are coefficients and constants and o is also a constant. It can also be observed that the power in the given expression is not a negative integer.
Examples of polynomial expressions
The following are algebraic examples of polynomial expressions:
2x³ - 4y + 5
y³ + 3x² - x + 2
x³ + 2y² - 3y³ + y + 2
Practical example
A company produces a radio set, and the production cost (C) is dependent on the number of radios produced (Y). The production cost can be represented using a polynomial expression thus:
(C)x = 2y² + 12y + 42
where 2y² stands for the cost of materials, which will increase with the number of radios produced in a quadratic manner, 12y stands for the labor cost, which will increase with the number of radios produced in a linear manner, and 42 is the fixed production cost.
Rational Expressions
A rational expression is a fractional algebraic, mathematical expression that is composed of polynomial expressions. That is to say that it is a fraction in which the numerators and the denominators are all polynomial expressions.
Rational expressions are characterized by a fraction that has polynomials as both the numerator and the denominator, and variables that are contained in the polynomial numerators and denominators.
Generally, rational expressions take the form of:
a(x) / b(x)
where x is the variable, and a and b are the polynomial numerator and denominator, respectively.
Examples of Rational expressions
Algebraic examples of rational expressions include the following:
(2y² + 3y + 2) / (y² - 2)
(x² - 2x + 2) / (x + 2)
(3y + 2) / (3 - y)
Practical example
A company produces a radio set, and the production cost (C) is dependent on the number of radios produced (Y). The cost function can be represented using a rational expression thus:
(C)x = (3y + 150) / (y + 4)
where 3y indicates the variable production cost, 150 is the fixed production cost, y + 4 is the total number of radios produced, and 4 is the minimum.
Differences between polynomials and rational expressions.
| Definition | A polynomial expression refers to an algebraic mathematical expression that is made up of variables, arithmetic operations as well as coefficients in which the index or power of the variables are not negative integers | A rational expression is a fractional algebraic mathematical expression that is composed of polynomial expressions. That is to say that it is a fraction in which the numerators and the denominators are all polynomial expressions |
| General form | mx² + ny - o | a(x) / b(x) |
| Powers and variables | Variables may have indices that are not negative | Variables may have negative indices |
| Operation | Can be added, divided, multiplied, and subtracted | Can be added, multiplied, and subtracted; division can be done only when the denominator is not zero |
| Simplification | Done by grouping like terms | Done by cancelling common factors in the fraction |
Explain steps used in simplifying a rational expression. Write some common factors required to be cancel out?
Although simplifying a rational expression calls for canceling the common factors in both the numerator and the denominator, several steps are required and they include:
Step 1: Factoring
This is the first step required for simplifying a rational expression. The numerator and the denominator should first be factored into their various prime factors.
Step 2: Spotting the common factors
Factoring the numerator and the denominator into their prime factors makes them expanded giving room for the next step which is identifying available common factor(s)
Step 3: Cancelling common factors
In this step, common factor(s) that exist between the numerator and the denominator are canceled out any comm
Step 4: Simplification
This is the final step, and here whatever is left after canceling the common factors is simplified where possible.
Example
(3a² - 6ab) / (2a²b - 4ab²)
Solution
Step 1: Factoring the expression
(3a² - 6ab) / (2a²b - 4ab²)
= (3a(a - 2b)) / (2ab(a - 2b)
Step 2: Identify common factors
The common factor in the numerator and denominator is (a - 2b)
Step 3: Cancelling common factors
By canceling the common factors in the expression we will have
(3a(a - 2b)) / (2ab(a - 2b)
= 3a/2ab
Step 3: Simplifying
3a/2ab
= (3 × a) / (2 × a × b) ===> by expansion
a will be canceled out
= 3/2b
Therefore, (3a² - 6ab) / (2a²b - 4ab²) = 3/2b
Please add these polynomial expressions 3x^2 + 2x + 1 and 2x^2 - x - 3 and share your final expression.
Adding the expressions 3x² + 2x + 1 and 2x² - x - 3
We will have
(3x² + 2x + 1) + (2x² - x - 3)
Clearing the brackets well will have
3x² + 2x + 1 + 2x² - x - 3
The next step will be regrouping or collecting like terms and then simplify
3x² + 2x² + 2x - x + 1 - 3
5x² + x - 2
Therefore, adding the expressions 3x² + 2x + 1 and 2x² - x - 3, we will have 5x² + x - 2
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)
(2x² + 3)(x - 2)
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 scenario based questions and share step by step that how you reach to your final outcome)
Step 1: This will be to determine the original cost before the percentage discount, so
(C)x = 5 + 2x
(this encompasses the $5 fixed cost and $2 variable cost for each packet)
Step 2: We will here, add the 10% discount which will require us to find 10% of the initial price and subtract it from the initial price as well
Discount of 10% = 10/100 = 0.1 × (C)x
= 0.1 × (5 + 2x)
= 0.5 + 0.2x
Next, we subtract this from the initial cost of (C)x = 5 + 2x and simplify
(C)x = (5 + 2x) - (0.5 + 0.2x)
Clearing the brackets, collecting like terms, and simplifying
= 5 + 2x - 0.5 - 0.2x
= 5 - 0.5 + 2x - 0.2x
= 4.5 + 1.8x
Step 3: This will be to calculate the cost of purchasing 5 packets of beads, that is x = 5
C = 4.5 + 1.8x
= 4.5 + 1.8(5)
= 4.5 + 9
= 13.5
Therefore, the final outcome is $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.
In this scenario, the first thing we have to do is to write a rational expression for representing ratio of wheat to total harvest, thus, the ratio of wheat to total harvest will be
Wheat : total harvest = wheat / total harvest
= x / (x + 3x)
The next thing will be to simplify the expression by adding the like terms of the denominator and canceling out common factors
= x /4x
= 1/4
Having done that, we proceed to calculate the ratio of wheat to total harvest if the farmer is harvesting 4 tons of wheat
Total harvest = x + 3x = 4x
Where x = 4 tons(of wheat), total harvest will be
4x = 4(4)
= 16
Now, we calculate the ratio of wheat to total harvest
= wheat / total harvest
= 4/16
= 1/4 or 1:4
Thank you
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