Subtraction of Rational Expressions

Subtracting rational expressions involves finding a common denominator and subtracting the numerators.

It requires ensuring the denominators are the same before performing the subtraction on the numerators.

There are three components of a rational expression:

Numerator: This contains constants, variables, or both.

Denominator: This also contains constants, variables, or both and must not be zero.

Operators: For subtraction, the operator is the minus (-) symbol.

When subtracting rational expressions, students should follow these steps:

Find a common denominator: The expressions must have the same denominator to subtract them.

Subtract numerators: Once the denominators are the same, subtract the numerators while keeping the common denominator.

Simplify the result: Simplify the resulting rational expression by combining like terms and reducing the fraction if possible.

The following are the methods of subtraction of rational expressions:

Method 1: Common Denominator Method

To apply the common denominator method for subtraction of rational expressions, use the following steps.

Step 1: Identify the least common denominator (LCD) of the expressions.

Step 2: Rewrite each expression with the LCD as the denominator.

Step 3: Subtract the numerators.

Let’s apply these steps to an example:

Question: Subtract 1/x-2  from 2/x+3

Step 1: Find the LCD, which is (x-2)(x+3).

Step 2: Rewrite each expression with the common denominator.

Step 3: Subtract the numerators.

Answer: (2)(x-2) - (1)(x+3)/(x-2)(x+3)

Method 2: Simplification Method

When subtracting the rational expressions using the simplification method, simplify each expression before performing subtraction.

If possible, factor and reduce the expressions to their simplest form before finding a common denominator.

For example, Subtract x²/x²-1 from 2x/x-1

Solution: Simplify each expression first, then find the common denominator, (x-1)(x+1).

x²/(x-1)(x+1) - 2x(x+1)/(x-1)(x+1) = x² - 2x(x+1)/(x-1)(x+1)

Therefore, upon subtracting, we get x² - 2x² - 2x/(x-1)(x+1).

In rational expressions, subtraction has some characteristic properties.

These properties are listed below:

Subtraction is not commutative In subtraction, changing the order of the terms changes the result, i.e., A/B - C/D neq C/D - A/B

Subtraction is not associative Unlike addition, we cannot regroup in subtraction.

When three or more expressions are involved, changing the grouping changes the result. A/B - C/D - E/F neq A/B - C/D - E/F

Subtraction is the addition of the opposite sign Subtracting an expression is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second term.

A/B - C/D = A/B + (-C/D)

Subtracting zero from an expression leaves the expression as is Subtracting zero from any expression results in the same rational expression: A/B - 0 = A/B

Tips and tricks are useful for students to efficiently deal with the subtraction of rational expressions. Some helpful tips are listed below:

Tip 1: Always pay attention to signs before combining numerators.

Tip 2: Ensure the denominators are the same before subtracting. This makes the expressions easier to handle and provides more clarity.

Tip 3: Beginners and visual learners can benefit from factoring and simplifying expressions first to avoid complexity.

Subtraction in rational expressions is often more challenging than addition, leading to common mistakes. However, being aware of these errors can help students avoid them.