Addition And Subtraction Of Rational Expressions With Unlike Denominators

Adding and subtracting rational expressions with unlike denominators requires a solid understanding of fractions, factoring, and algebraic manipulation. Just like adding or subtracting numerical fractions, you must first find a common denominator before combining the numerators. This article will provide a comprehensive guide to mastering this essential skill in algebra.

Understanding Rational Expressions

A rational expression is simply a fraction where the numerator and denominator are polynomials. Examples include:

• (x + 2) / (x - 3)
• (2x^2 - 1) / (x + 5)
• 5 / (x^2 + 2x + 1)

The key challenge when adding or subtracting these expressions arises when they have different denominators. You can't directly combine the numerators until you establish a common denominator. This process involves finding the Least Common Denominator (LCD) and rewriting each fraction with this new denominator.

Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that is divisible by all the denominators in the problem. Here's a step-by-step guide to finding the LCD:

1. Factor each denominator completely. This means breaking down each polynomial into its prime factors. For example:
   • x^2 - 4 = (x + 2)(x - 2)
   • x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2
2. Identify all unique factors. List all the different factors that appear in any of the denominators. For example, if you have denominators of (x + 1), (x - 2), and (x + 1)(x + 3), your unique factors are (x + 1), (x - 2), and (x + 3).
3. Determine the highest power of each unique factor. For each unique factor, find the highest power to which it appears in any of the denominators.
   • If one denominator has (x + 2) and another has (x + 2)^2, the highest power is (x + 2)^2.
4. Multiply the highest powers of all unique factors. The product of these highest powers is the LCD.

Example 1:

Find the LCD of:

• 1 / (x + 1)
• 1 / (x - 1)

1. Factor: Both denominators are already in their simplest factored form.
2. Unique Factors: (x + 1) and (x - 1)
3. Highest Power: Each factor appears to the first power.
4. LCD: (x + 1)(x - 1)

Example 2:

Find the LCD of:

• 1 / (x^2 - 4)
• 1 / (x + 2)

1. Factor:
   • x^2 - 4 = (x + 2)(x - 2)
   • x + 2 = (x + 2)
2. Unique Factors: (x + 2) and (x - 2)
3. Highest Power: (x + 2) appears to the first power in the second denominator and to the first power within the factored form of the first denominator. (x - 2) appears to the first power.
4. LCD: (x + 2)(x - 2) which can also be written as x^2 - 4

Example 3:

Find the LCD of:

• 1 / (x^2 + 5x + 6)
• 1 / (x^2 + 6x + 9)

1. Factor:
   • x^2 + 5x + 6 = (x + 2)(x + 3)
   • x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
2. Unique Factors: (x + 2) and (x + 3)
3. Highest Power: (x + 2) appears to the first power. (x + 3) appears to the second power.
4. LCD: (x + 2)(x + 3)^2

Adding and Subtracting Rational Expressions: Step-by-Step

Here's the process of adding and subtracting rational expressions with unlike denominators:

1. Find the LCD. As described above, factor all denominators completely and determine the LCD.
2. Rewrite each fraction with the LCD as the denominator. Multiply the numerator and denominator of each fraction by the factors needed to make the denominator equal to the LCD. This is crucial – you're essentially multiplying by "1" in a strategic way to change the form of the fraction without changing its value.
3. Add or subtract the numerators. Once all fractions have the same denominator, you can combine the numerators. Remember to pay attention to the signs, especially when subtracting.
4. Simplify the resulting expression.
   • Combine like terms in the numerator.
   • Factor the numerator and denominator (if possible) to see if any factors can be canceled.
   • State any restrictions on the variable (values that would make the denominator zero).

Example 1: Addition

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

1. LCD: (x + 1)(x - 2)

2. Rewrite Fractions:

   • (2 / (x + 1)) * ((x - 2) / (x - 2)) = (2(x - 2)) / ((x + 1)(x - 2))
   • (3 / (x - 2)) * ((x + 1) / (x + 1)) = (3(x + 1)) / ((x + 1)(x - 2))

3. Add Numerators:

   • (2(x - 2) + 3(x + 1)) / ((x + 1)(x - 2)) = (2x - 4 + 3x + 3) / ((x + 1)(x - 2))

4. Simplify:

   • (5x - 1) / ((x + 1)(x - 2))

   This expression is already simplified, and no further factoring is possible.

Example 2: Subtraction

Subtract: (x / (x - 3)) - (2 / (x + 4))

1. LCD: (x - 3)(x + 4)

2. Rewrite Fractions:

   • (x / (x - 3)) * ((x + 4) / (x + 4)) = (x(x + 4)) / ((x - 3)(x + 4))
   • (2 / (x + 4)) * ((x - 3) / (x - 3)) = (2(x - 3)) / ((x - 3)(x + 4))

3. Subtract Numerators:

   • (x(x + 4) - 2(x - 3)) / ((x - 3)(x + 4)) = (x^2 + 4x - 2x + 6) / ((x - 3)(x + 4))

4. Simplify:

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

   This expression is already simplified, and no further factoring is possible.

Example 3: With Factoring Needed

Subtract: (3 / (x^2 - 1)) - (1 / (x + 1))

1. LCD:
   • Factor: x^2 - 1 = (x + 1)(x - 1)
   • LCD: (x + 1)(x - 1)
2. Rewrite Fractions:
   • (3 / ((x + 1)(x - 1))) - (1 / (x + 1)) * ((x - 1) / (x - 1)) = (3 / ((x + 1)(x - 1))) - ((x - 1) / ((x + 1)(x - 1)))
3. Subtract Numerators:
   • (3 - (x - 1)) / ((x + 1)(x - 1)) = (3 - x + 1) / ((x + 1)(x - 1))
4. Simplify:
   • (4 - x) / ((x + 1)(x - 1))

Example 4: Simplifying After Combining

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

1. LCD:

   • Factor: x^2 - 4 = (x + 2)(x - 2)
   • LCD: (x + 2)(x - 2)

2. Rewrite Fractions:

   • (x / ((x + 2)(x - 2))) + (2 / (x - 2)) * ((x + 2) / (x + 2)) = (x / ((x + 2)(x - 2))) + (2(x + 2) / ((x + 2)(x - 2)))

3. Add Numerators:

   • (x + 2(x + 2)) / ((x + 2)(x - 2)) = (x + 2x + 4) / ((x + 2)(x - 2))

4. Simplify:

   • (3x + 4) / ((x + 2)(x - 2))

   This expression is already simplified, and no further factoring is possible.

Example 5: More Complex Factoring

Add: (2x / (x^2 + 3x + 2)) + ((x - 1) / (x^2 + 4x + 3))

1. LCD:

   • Factor: x^2 + 3x + 2 = (x + 1)(x + 2)
   • Factor: x^2 + 4x + 3 = (x + 1)(x + 3)
   • LCD: (x + 1)(x + 2)(x + 3)

2. Rewrite Fractions:

   • (2x / ((x + 1)(x + 2))) * ((x + 3) / (x + 3)) = (2x(x + 3)) / ((x + 1)(x + 2)(x + 3))
   • ((x - 1) / ((x + 1)(x + 3))) * ((x + 2) / (x + 2)) = ((x - 1)(x + 2)) / ((x + 1)(x + 2)(x + 3))

3. Add Numerators:

   • (2x(x + 3) + (x - 1)(x + 2)) / ((x + 1)(x + 2)(x + 3)) = (2x^2 + 6x + x^2 + x - 2) / ((x + 1)(x + 2)(x + 3))

4. Simplify:

   • (3x^2 + 7x - 2) / ((x + 1)(x + 2)(x + 3))

   It is often difficult to factor the resulting quadratic in the numerator, so often it will not factor and simplify.

Common Mistakes to Avoid

• Forgetting to distribute the negative sign when subtracting. This is a very common error. Always treat the entire numerator being subtracted as a single unit and distribute the negative sign to every term within it.
• Incorrectly factoring denominators. Make sure you factor denominators completely and accurately before finding the LCD. A mistake in factoring will lead to an incorrect LCD and an incorrect final answer.
• Canceling terms before finding a common denominator. You can only cancel common factors from the numerator and denominator of a single fraction. You cannot cancel terms across different fractions being added or subtracted.
• Not simplifying the final answer. Always look for opportunities to factor and cancel to simplify the expression.
• Ignoring restrictions on the variable. Remember that the denominator of a fraction cannot be zero. Identify any values of the variable that would make any of the original denominators zero and exclude them from the solution.

Restrictions on the Variable

A critical aspect of working with rational expressions is identifying any values of the variable that would make the denominator equal to zero. These values are called restrictions and must be excluded from the domain of the expression. To find the restrictions, set each denominator equal to zero and solve for the variable.

Example:

Consider the expression: (x + 1) / (x - 2)

To find the restriction, set the denominator equal to zero:

x - 2 = 0 x = 2

Therefore, x cannot equal 2. This restriction should be noted alongside the simplified expression, often written as:

(x + 1) / (x - 2), x ≠ 2

In the earlier examples, restrictions should also be considered.

• In Example 1, (5x - 1) / ((x + 1)(x - 2)), x ≠ -1, 2
• In Example 2, (x^2 + 2x + 6) / ((x - 3)(x + 4)), x ≠ 3, -4
• In Example 3, (4 - x) / ((x + 1)(x - 1)), x ≠ -1, 1
• In Example 4, (3x + 4) / ((x + 2)(x - 2)), x ≠ -2, 2
• In Example 5, (3x^2 + 7x - 2) / ((x + 1)(x + 2)(x + 3)), x ≠ -1, -2, -3

Practice Problems

Here are some practice problems to help you solidify your understanding. Work through each problem carefully, paying attention to each step.

1. (3 / (x + 2)) + (4 / (x - 1))
2. (x / (x - 5)) - (2 / (x + 3))
3. (5 / (x^2 - 9)) + (1 / (x + 3))
4. (2x / (x^2 + 4x + 3)) - (x / (x + 1))
5. (x + 1) / (x^2 - 2x - 8) + (2 / (x - 4))

Answers:

1. (7x + 5) / ((x + 2)(x - 1)), x ≠ -2, 1
2. (x^2 + x + 10) / ((x - 5)(x + 3)), x ≠ 5, -3
3. (x + 8) / ((x + 3)(x - 3)), x ≠ -3, 3
4. (-x^2 + x) / ((x + 1)(x + 3)), x ≠ -1, -3
5. (3x + 5) / ((x - 4)(x + 2)), x ≠ 4, -2

Conclusion

Adding and subtracting rational expressions with unlike denominators is a fundamental skill in algebra. By mastering the process of finding the LCD, rewriting fractions, and simplifying the results, you'll be well-equipped to tackle more complex algebraic problems. Remember to pay attention to detail, avoid common mistakes, and practice consistently to build your confidence and proficiency. Always double-check your work and remember to state the restrictions on the variable to ensure a complete and accurate solution. With dedication and practice, you can successfully navigate the challenges of rational expressions and excel in your algebra studies.