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how do rational expressions connect to polynomials?

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how do rational expressions connect to polynomials?

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The definition of a polynomial is  an expression that has variables that normally are involved with adding subtracting multiplying or dividing. An example is shown below. Anyways, the way polynomials connect to rational expressions are the fact that rational expressions use polynomials for both the numerator and denominator!

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These tables show values plugged into the equation and their outputs. In example No. 1 and No. 2 we plugged the same values into each equation to see if they would give the same answer and they did; this means that these two expressions are equal.

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Graphing Example: No. 2 - Multiplication 

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Graphing Example: No. 2 - Multiplication 

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Click play!
The video below shows that the two expressions are equivalent. The first expression, the red curve, is the original expression, while the second expression is the simplified first expression.
Based on the points in the graph you can see in the graph, it's obvious that the points line up and the lines overlap.

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Multiplying rational expressions by graphing

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In this algebraic example, we solved the original equation to see if the answer was equal to the simplified answer, since the simplified version didn't have any variables. In this example, when we plugged in x=3, we found that the original equation was equal to the simplified version.

The problem                                        The answer

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Click play!
This video shows how the two expressions are equal. The first expression, the red curves, is the original equation, while the second expression, the blue curves, is the first equation, but simplified.
From the points that are shown on the graph, you can see that the two expressions are equivalent! The x value and y values are the same.

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In this algebraic example, we solved the original equation and the simplified answer with the same x value; we did this to see if they would have the same answer, or, equal each other. In this example, when we plugged in x=2, we found that the two equation's answers were equal;
therefore, by extension, the two expressions are equal.

The problem                                           The answer (simplified)

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Solving Rational Expressions using a table (advantages and disadvantages)

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Solving Rational Expressions using a table (advantages and disadvantages)

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Advantages: The advantages to making tables is that it’s a clear way to see how the expressions are equal. It shows the input and output in the most organized way.

Disadvantages: Using a table doesn’t allow the student to completely understand the work or the formula that is attached to it. The student can just plug in numbers to the expression and find the outputs.

Disadvantage: Graphing can easily replace this method. Graphing not only shows you the shape, but can also show you the input and outputs, taking away from the importance of using a table

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Algebraically Simplifying  Rational Expressions (Advantages and Disadvantages)

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Algebraically Simplifying  Rational Expressions (Advantages and Disadvantages)

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Advantage: Showing all of the work can help you see where the work was wrong and it can easily be fixed. This is the most valuable method to use! It never goes wrong.

Advantage: Its straightforward

Disadvantage: The only disadvantage of this method is human error. If a problem is done wrong, than the two expressions wouldn’t be equivalent.

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Modeling Rational Expressions (Advantages and  Disadvantages)

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Modeling Rational Expressions (Advantages and  Disadvantages)

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Advantage: This shows a Step-by-Step on how to simplify an expression. It helps to see the base and the starting point to the ending point with all of your work shown. This can help prove that you understand the work given

Advantage: Helps visual learners connect to math easier

Advantage: Makes rational expressions more visually appealing

Disadvantage: The disadvantage of modeling is the fact that you can see so many steps going on to a point where you have no idea what's going on.

Disadvantage: It might become really messy if not used carefully

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Graphing Rational Expressions (Advantages and  Disadvantages)

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Graphing Rational Expressions (Advantages and  Disadvantages)

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Advantage: Graphing helps to visualize the expressions that are being compared. It helps you easily look for the coordinate points and see the overall shape of the equation. On Desmos, the two expressions perfectly line up if they’re equal, so that’s an easy way to envision the expressions

Advantage: Graphing shows you all the coordinate points so you can see all the minor points and how they connect. It even shows you the asymptote in some cases, which is the number that will get you 0 as your simplified answer

Disadvantage: The disadvantage of this is that people will become reliant on graphing and won’t understand how the math actually works. They will know that the two expressions are equal, but they won’t know how it worked algebraically or how to prove that another way.

Another Disadvantages is the fact that the graphs move out indefinitely, for infinity.  Which means, at some point, where we cant see on the graph, there is a possibiity that they are not rqual

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Steps for Subtracting Rational Expressions 

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Steps for Subtracting Rational Expressions 

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1. Factor where applicable.

2. Find a common denominator.

1. In this case, we found a common denominator by multiplying the first term by (x+5)

3. Rewrite problem using the new common denominator/s.

4. Subtract the numerators and keep the denominator the same.

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1. Factor where applicable.

2. Find a common denominator

1. To find a common denominator,  you have to multiply the denominators by each other. Note: If one factor is already present, such as (x-4) in the example, than you only need to multiply by the other factor. In the example, we only multiplied the second term by the part of the denominator that was missing (x+2), in this step.

3. Rewrite problem using the new common denominator/s

4. Add the numerators and keep the denominator the same

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1. Keep the first fraction, change the division sign to a multiplication sign, and write the reciprocal of the second fraction

1. Note: Switching the sign and finding the reciprocal has the same effect on the equation has multiplying the first term by the numerator of the second term and dividing that by the denominator of the second term; which is why this method works.

2. By using the commutative property, rearrange the terms so that like terms are near each other

3. Simplify.

4. Simplify further.

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Steps for Multiplying Rational Expressions

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Steps for Multiplying Rational Expressions

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1. If applicable, put the standard form expressions into factored form

2. Rearrange the terms to take out the unnecessary ones (use the commutative property)

1. Note: by taking out the unnecessary ones, you are using the multiplicative identity property. The multiplicative identity is the property that allows numbers to be multiplied by one and not have it's values changed. (ex: 9*1=9)

3. Multiply the numerators and denominators

4. Simplify if needed

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Similarities Between Rational Expressions and Rational Numbers

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Similarities Between Rational Expressions and Rational Numbers

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Rational Numbers are whole numbers, or any number that isn’t a decimal and can’t be turned into one (fractions). Rational expressions have terms, variables and rational numbers, while rational numbers are just the number itself. In addition, rational expressions can be simplified, while rational numbers are already in their simplest form. It’s important to remember that rational expressions are just rational numbers put into expressions.

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Rational Expressions are a form of a polynomial that combines both rational numbers and variables and is placed in the format of an expression (no equal sign). These expressions can be in standard (ex: ax² + bx + c) or in factored form (x+r)(x+s). Rational expressions can’t be solved but they can be simplified.

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A rational number is any number that can be represented as a fraction that has a non zero denominator.