Adding / Subtracting Rational Expressions

Mastering the Art of Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions might seem daunting at first, but with a systematic approach and a solid understanding of fractions, it becomes a manageable and even enjoyable mathematical skill. This comprehensive guide will walk you through the process step-by-step, providing clear explanations, examples, and helpful tips to master this crucial algebra concept. Whether you're a high school student tackling algebra or an adult brushing up on your math skills, this guide will equip you with the confidence to tackle any rational expression problem.

Understanding Rational Expressions

Before diving into addition and subtraction, let's clarify what a rational expression is. Simply put, a rational expression is a fraction where the numerator and/or the denominator are polynomials. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

For example, these are all rational expressions:

• 3x / (x² + 2)
• (x² - 4) / (x + 1)
• (2x³ + 5x - 1) / (x⁴ - 3)

Remember, just like with regular fractions, the denominator cannot be equal to zero. This means we must always consider the values of the variable(s) that would make the denominator zero, and exclude them from the domain of the rational expression.

Adding and Subtracting Rational Expressions with Common Denominators

Adding and subtracting rational expressions with common denominators is the simplest case. It mirrors the process of adding and subtracting ordinary fractions.

Rule: If two rational expressions have the same denominator, add or subtract their numerators and keep the common denominator. Then, simplify the resulting expression if possible.

Example 1: Adding

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

Since both expressions have the denominator (x - 3), we add the numerators:

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

So the result is: (3x - 3) / (x - 3)

This expression can be simplified further by factoring the numerator:

(3(x - 1)) / (x - 3)

Example 2: Subtracting

(5x² + 2x) / (x² + 1) - (3x² - x) / (x² + 1)

Subtracting the numerators while keeping the common denominator:

(5x² + 2x) - (3x² - x) = 5x² + 2x - 3x² + x = 2x² + 3x

The simplified result is: (2x² + 3x) / (x² + 1)

Finding the Least Common Denominator (LCD)

Adding and subtracting rational expressions with different denominators requires finding the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators in the problem.

Steps to find the LCD:

1. Factor each denominator completely: Express each denominator as a product of its prime factors (both numerical and polynomial factors).

2. Identify the unique factors: List all the unique factors found in the denominators.

3. Determine the highest power of each factor: For each unique factor, choose the highest power that appears in any of the denominators.

4. Multiply the highest powers: The LCD is the product of the highest powers of all the unique factors.

Example: Find the LCD of (3/ (x² - 4)) and (2x/ (x² - 5x + 6))

1. Factor the denominators:

   • x² - 4 = (x - 2)(x + 2)
   • x² - 5x + 6 = (x - 2)(x - 3)

2. Unique factors: (x - 2), (x + 2), (x - 3)

3. Highest powers: Each factor appears only to the first power.

4. LCD: (x - 2)(x + 2)(x - 3)

Adding and Subtracting Rational Expressions with Different Denominators

Once you've found the LCD, you need to rewrite each rational expression with the LCD as the denominator. This involves multiplying both the numerator and the denominator of each expression by the appropriate factors. Remember, multiplying both the numerator and denominator by the same factor doesn't change the value of the expression (it's equivalent to multiplying by 1).

Steps:

1. Find the LCD: Follow the steps outlined above.

2. Rewrite each expression with the LCD: Multiply the numerator and denominator of each expression by the factors needed to obtain the LCD.

3. Add or subtract the numerators: Keep the common denominator.

4. Simplify: Factor the numerator and cancel out any common factors between the numerator and denominator.

Example: Add (3/ (x² - 4)) + (2x/ (x² - 5x + 6))

1. LCD: (x - 2)(x + 2)(x - 3) (from the previous example)

2. Rewrite with LCD:

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

3. Add numerators:

   • [3(x - 3) + 2x(x + 2)] / ((x - 2)(x + 2)(x - 3)) = (3x - 9 + 2x² + 4x) / ((x - 2)(x + 2)(x - 3)) = (2x² + 7x - 9) / ((x - 2)(x + 2)(x - 3))

4. Simplify (if possible): In this case, the numerator doesn't factor nicely, so the expression is simplified.

Example with subtraction:

Subtract (x/(x+2)) - (2/(x-1)).

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

2. Rewrite:

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

3. Subtract Numerators:

   • [x(x-1) - 2(x+2)]/((x+2)(x-1)) = (x² - x - 2x - 4)/((x+2)(x-1)) = (x² -3x -4)/((x+2)(x-1))

4. Simplify: We can factor the numerator: (x-4)(x+1)/((x+2)(x-1))

Dealing with Complex Polynomials

When dealing with more complex polynomials, factoring becomes critical. Remember the different factoring techniques like:

• Greatest Common Factor (GCF): Factoring out the largest common factor from all terms.
• Difference of Squares: a² - b² = (a + b)(a - b)
• Perfect Square Trinomial: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
• Factoring Trinomials: Using techniques like the AC method or grouping to factor quadratic trinomials.

Mastering these factoring techniques significantly improves your ability to find LCDs and simplify rational expressions.

Simplifying the Result

After performing the addition or subtraction, always simplify the resulting expression as much as possible. This involves:

• Factoring: Factor both the numerator and the denominator to identify common factors.

• Cancelling Common Factors: Cancel out any common factors that appear in both the numerator and the denominator.

Remember that you can only cancel factors, not terms.

Frequently Asked Questions (FAQ)

Q: What if I get a complex fraction after adding or subtracting?

A: A complex fraction is a fraction where the numerator or denominator (or both) contains a fraction. To simplify, treat it as a division problem. Multiply the numerator by the reciprocal of the denominator.

Q: Can I add or subtract rational expressions with different variables?

A: Yes, but finding the LCD might be more challenging. You'll need to consider all the variable parts in the denominators when determining the LCD. For example, if you have denominators involving 'x' and 'y', your LCD will likely contain both 'x' and 'y' terms.

Q: What if I make a mistake in factoring?

A: Carefully review your factoring steps. Use techniques like checking your factoring by multiplying the factors back together to see if you obtain the original polynomial.

Q: How do I know when my answer is simplified?

A: Your answer is generally simplified when there are no common factors between the numerator and the denominator, and the numerator and denominator are in simplest polynomial form.

Conclusion

Adding and subtracting rational expressions is a fundamental skill in algebra. By understanding the concepts of common denominators, LCDs, and factoring, and by following the steps outlined in this guide, you can confidently tackle even the most challenging problems. Practice is key; the more you work through examples, the more comfortable you'll become with this essential mathematical process. Remember to always check your work and simplify your answers as much as possible. With dedication and perseverance, mastering rational expressions will significantly enhance your algebraic abilities.