How do you add polynomial fractions?

To add polynomial fractions, you first need to find a common denominator, then combine the numerators. Let's break down the process:

1. Find a Common Denominator

• Factor each denominator completely. This is crucial to identify common factors.
• Identify the Least Common Denominator (LCD). The LCD is the smallest expression that is divisible by each of the original denominators. It's built by including all unique factors from all denominators, raised to the highest power they appear in any single denominator.

2. Rewrite Each Fraction with the Common Denominator

• Multiply the numerator and denominator of each fraction by the necessary factors to obtain the LCD as the new denominator. This ensures that the value of each fraction remains unchanged.

3. Add the Numerators

• Once all fractions have the same denominator, add their numerators. Keep the common denominator. This is similar to adding regular fractions.

4. Simplify the Result

• Combine like terms in the numerator.
• Factor both the numerator and the denominator.
• Cancel any common factors between the numerator and the denominator to simplify the fraction to its lowest terms.

Example

Let's add the following polynomial fractions:

(x + 1) / (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 fraction with the LCD:

   • (x + 1) / (x^2 - 4) = (x + 1) / ((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. Add the numerators:

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

5. Simplify the result:

   • Factor the numerator: (3(x - 1)) / ((x + 2)(x - 2))
   • In this case, there are no common factors to cancel.

Therefore, the simplified answer is (3(x - 1)) / ((x + 2)(x - 2)).