Adding And Subtracting Rational Expressions Examples

Rational expressions, often perceived as complex algebraic fractions, become manageable when approached systematically. Adding and subtracting these expressions is a fundamental skill in algebra, crucial for simplifying equations and solving real-world problems. This comprehensive guide breaks down the process with clear explanations and practical examples.

Understanding Rational Expressions

A rational expression is simply a fraction where the numerator and denominator are polynomials. Examples include (x+1)/(x-2) and (3x^2 - 2x + 5)/(x+3). Just like numerical fractions, rational expressions can be added, subtracted, multiplied, and divided. The key to adding and subtracting them lies in finding a common denominator.

Key Concepts:

• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Fractions: Expressions representing a part of a whole, consisting of a numerator and a denominator.
• Common Denominator: A denominator that is the same for two or more fractions, allowing for addition or subtraction.

The Foundation: Numerical Fractions

Before diving into rational expressions, let's revisit how to add and subtract numerical fractions. This will solidify the underlying principles that apply to algebraic fractions as well.

Adding Fractions:

To add fractions like 1/3 + 1/4, we need a common denominator. The least common multiple (LCM) of 3 and 4 is 12. We then rewrite each fraction with the denominator of 12:

• 1/3 = (1 * 4) / (3 * 4) = 4/12
• 1/4 = (1 * 3) / (4 * 3) = 3/12

Now we can add:

• 4/12 + 3/12 = (4 + 3) / 12 = 7/12

Subtracting Fractions:

The process is similar for subtraction. For example, to subtract 2/5 - 1/3, we again find the LCM of 5 and 3, which is 15. We rewrite the fractions:

• 2/5 = (2 * 3) / (5 * 3) = 6/15
• 1/3 = (1 * 5) / (3 * 5) = 5/15

Now we subtract:

• 6/15 - 5/15 = (6 - 5) / 15 = 1/15

Steps to Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions involves a few key steps:

1. Factor the Denominators: Factor each denominator completely. This is crucial for identifying the least common denominator (LCD).
2. Find the Least Common Denominator (LCD): The LCD is the smallest expression that is divisible by each of the original denominators. To find it, include each factor that appears in any denominator, raised to the highest power it appears in any one denominator.
3. Rewrite Each Rational Expression: Multiply the numerator and denominator of each rational expression by the factors needed to make the denominator equal to the LCD.
4. Add or Subtract the Numerators: Once all the expressions have the same denominator, you can add or subtract the numerators. Keep the common denominator.
5. Simplify the Result: Simplify the resulting rational expression by factoring the numerator and denominator and canceling any common factors.

Example 1: Simple Denominators

Let's start with a straightforward example:

(x/3) + (2x/5)

1. Factor the Denominators: The denominators 3 and 5 are already in their simplest form.
2. Find the LCD: The LCD of 3 and 5 is 15.
3. Rewrite Each Rational Expression:
   • (x/3) = (x * 5) / (3 * 5) = 5x/15
   • (2x/5) = (2x * 3) / (5 * 3) = 6x/15
4. Add the Numerators:
   • (5x/15) + (6x/15) = (5x + 6x) / 15 = 11x/15
5. Simplify the Result: The expression 11x/15 is already in its simplest form.

Therefore, (x/3) + (2x/5) = 11x/15.

Example 2: Denominators with Variables

Consider the following:

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

1. Factor the Denominators: The denominators (x+1) and (x-2) are already factored.
2. Find the LCD: The LCD is (x+1)(x-2).
3. Rewrite Each Rational Expression:
   • (3/(x+1)) = (3 * (x-2)) / ((x+1) * (x-2)) = (3x - 6) / (x^2 - x - 2)
   • (2/(x-2)) = (2 * (x+1)) / ((x-2) * (x+1)) = (2x + 2) / (x^2 - x - 2)
4. Add the Numerators:
   • ((3x - 6) / (x^2 - x - 2)) + ((2x + 2) / (x^2 - x - 2)) = (3x - 6 + 2x + 2) / (x^2 - x - 2) = (5x - 4) / (x^2 - x - 2)
5. Simplify the Result: The numerator (5x-4) cannot be factored further, and it shares no common factors with the denominator (x^2 - x - 2). Thus, the expression is in its simplest form.

Therefore, (3/(x+1)) + (2/(x-2)) = (5x - 4) / (x^2 - x - 2).

Example 3: Factoring Required

Let's tackle a problem where factoring is necessary:

(x/(x^2 - 4)) - (2/(x+2))

1. Factor the Denominators:
   • x^2 - 4 = (x + 2)(x - 2)
   • (x + 2) is already factored.
2. Find the LCD: The LCD is (x + 2)(x - 2).
3. Rewrite Each Rational Expression:
   • (x/(x^2 - 4)) = x / ((x + 2)(x - 2)) (already has the LCD)
   • (2/(x+2)) = (2 * (x - 2)) / ((x + 2) * (x - 2)) = (2x - 4) / ((x + 2)(x - 2))
4. Subtract the Numerators:
   • (x / ((x + 2)(x - 2))) - ((2x - 4) / ((x + 2)(x - 2))) = (x - (2x - 4)) / ((x + 2)(x - 2)) = (x - 2x + 4) / ((x + 2)(x - 2)) = (-x + 4) / ((x + 2)(x - 2))
5. Simplify the Result: We can rewrite the numerator as -(x-4) and the denominator as (x+2)(x-2). There are no common factors to cancel in this case. The result is (-x + 4) / ((x + 2)(x - 2)).

Therefore, (x/(x^2 - 4)) - (2/(x+2)) = (-x + 4) / ((x + 2)(x - 2)).

Example 4: More Complex Factoring

Consider this expression:

(4/(x^2 + 5x + 6)) + (2/(x^2 + 2x - 3))

1. Factor the Denominators:
   • x^2 + 5x + 6 = (x + 2)(x + 3)
   • x^2 + 2x - 3 = (x + 3)(x - 1)
2. Find the LCD: The LCD is (x + 2)(x + 3)(x - 1).
3. Rewrite Each Rational Expression:
   • (4/(x^2 + 5x + 6)) = (4 / ((x + 2)(x + 3))) = (4 * (x - 1)) / ((x + 2)(x + 3)(x - 1)) = (4x - 4) / ((x + 2)(x + 3)(x - 1))
   • (2/(x^2 + 2x - 3)) = (2 / ((x + 3)(x - 1))) = (2 * (x + 2)) / ((x + 3)(x - 1)(x + 2)) = (2x + 4) / ((x + 2)(x + 3)(x - 1))
4. Add the Numerators:
   • ((4x - 4) / ((x + 2)(x + 3)(x - 1))) + ((2x + 4) / ((x + 2)(x + 3)(x - 1))) = (4x - 4 + 2x + 4) / ((x + 2)(x + 3)(x - 1)) = (6x) / ((x + 2)(x + 3)(x - 1))
5. Simplify the Result: There are no common factors between 6x and the expanded form of (x + 2)(x + 3)(x - 1) which is x^3 + 4x^2 + x - 6. The result is (6x) / ((x + 2)(x + 3)(x - 1)).

Therefore, (4/(x^2 + 5x + 6)) + (2/(x^2 + 2x - 3)) = (6x) / ((x + 2)(x + 3)(x - 1)).

Example 5: Subtraction with a Negative Sign

Consider this subtraction problem:

(5/(x-4)) - (3/(4-x))

1. Factor the Denominators: Notice that (4-x) is almost the same as (x-4). We can rewrite (4-x) as -(x-4).
   • (x-4) remains as is.
   • (4-x) = -(x-4)
2. Find the LCD: The LCD is (x-4).
3. Rewrite Each Rational Expression:
   • (5/(x-4)) is already in the correct form.
   • (3/(4-x)) = (3/(-(x-4))) = (-3/(x-4))
4. Subtract the Numerators:
   • (5/(x-4)) - (-3/(x-4)) = (5 + 3)/(x-4) = 8/(x-4)
5. Simplify the Result: The expression 8/(x-4) is already in its simplest form.

Therefore, (5/(x-4)) - (3/(4-x)) = 8/(x-4).

Example 6: Combining Multiple Rational Expressions

Let's try adding and subtracting three rational expressions:

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

1. Factor the Denominators: The denominators x, (x+1), and (x-1) are already factored.
2. Find the LCD: The LCD is x(x+1)(x-1).
3. Rewrite Each Rational Expression:
   • (1/x) = ((x+1)(x-1)) / (x(x+1)(x-1)) = (x^2 - 1) / (x(x+1)(x-1))
   • (2/(x+1)) = (2x(x-1)) / (x(x+1)(x-1)) = (2x^2 - 2x) / (x(x+1)(x-1))
   • (3/(x-1)) = (3x(x+1)) / (x(x+1)(x-1)) = (3x^2 + 3x) / (x(x+1)(x-1))
4. Add and Subtract the Numerators:
   • ((x^2 - 1) / (x(x+1)(x-1))) + ((2x^2 - 2x) / (x(x+1)(x-1))) - ((3x^2 + 3x) / (x(x+1)(x-1))) = (x^2 - 1 + 2x^2 - 2x - 3x^2 - 3x) / (x(x+1)(x-1)) = (-5x - 1) / (x(x+1)(x-1))
5. Simplify the Result: The numerator (-5x-1) does not factor in a way that allows simplification with the denominator.

Therefore, (1/x) + (2/(x+1)) - (3/(x-1)) = (-5x - 1) / (x(x+1)(x-1)).

Example 7: An Advanced Example with Multiple Factors

(x+2)/(x^2 - 3x - 10) + (x-1)/(x^2 - 4)

1. Factor the Denominators:

   • x^2 - 3x - 10 = (x - 5)(x + 2)
   • x^2 - 4 = (x - 2)(x + 2)

2. Find the LCD: The LCD is (x - 5)(x + 2)(x - 2).

3. Rewrite Each Rational Expression:

   • (x+2)/((x - 5)(x + 2)) = ((x+2)(x-2)) / ((x - 5)(x + 2)(x-2)) = (x^2 - 4) / ((x - 5)(x + 2)(x-2))
   • (x-1)/((x - 2)(x + 2)) = ((x-1)(x-5)) / ((x - 2)(x + 2)(x-5)) = (x^2 - 6x + 5) / ((x - 5)(x + 2)(x-2))

4. Add the Numerators:

   • (x^2 - 4) / ((x - 5)(x + 2)(x-2)) + (x^2 - 6x + 5) / ((x - 5)(x + 2)(x-2)) = (x^2 - 4 + x^2 - 6x + 5) / ((x - 5)(x + 2)(x-2)) = (2x^2 - 6x + 1) / ((x - 5)(x + 2)(x-2))

5. Simplify the Result: The numerator (2x^2 - 6x + 1) does not factor easily and does not share factors with the denominator. Therefore, the expression is already simplified.

Therefore, (x+2)/(x^2 - 3x - 10) + (x-1)/(x^2 - 4) = (2x^2 - 6x + 1) / ((x - 5)(x + 2)(x-2)).

Common Mistakes to Avoid

• Forgetting to Factor: Always factor the denominators completely before finding the LCD.
• Incorrect LCD: Make sure the LCD includes all factors from each denominator, raised to the highest power they appear in any single denominator.
• Distributing Negatives: When subtracting, remember to distribute the negative sign to all terms in the numerator of the expression being subtracted.
• Incorrectly Simplifying: Only cancel common factors, not terms. For example, you cannot cancel the 'x' in (x+1)/x.

Practice Problems

To solidify your understanding, try these practice problems:

1. (2x/5) + (x/3)
2. (4/(x-2)) - (1/(x+2))
3. (x/(x^2 - 9)) + (3/(x+3))
4. (5/(x^2 + 2x + 1)) - (2/(x+1))
5. (1/(x+y)) + (1/(x-y))

Conclusion

Adding and subtracting rational expressions requires careful attention to detail, especially when factoring and finding the LCD. By following the steps outlined in this guide and practicing with examples, you can master this essential algebraic skill. Remember to always factor, find the LCD, rewrite the expressions, combine the numerators, and simplify the result. With consistent practice, these seemingly complex operations will become second nature.