nc math unit 3

To add or subtract polynomial fractions with unlike denominators, follow these steps:

1. Find a Common Denominator
   Determine the least common denominator (LCD) of the fractions. This usually involves factoring the denominators and taking the highest power of each unique factor present.

2. Rewrite Each Fraction
   Adjust each polynomial fraction so that they both have the common denominator.
   Multiply the numerator and denominator of each fraction by whatever polynomial(s) are needed to achieve the common denominator.

3. Perform the Addition or Subtraction
   Once both fractions have the same denominator, combine them by adding or subtracting the numerators while keeping the common denominator intact:

   \frac{A}{CD} + \frac{B}{CD} = \frac{A + B}{CD}

   or

   \frac{A}{CD} - \frac{B}{CD} = \frac{A - B}{CD}

4. Simplify the Result
   If possible, simplify the resulting fraction by factoring the numerator and reducing it by canceling out any common factors with the denominator.

To find a common denominator for polynomial fractions, follow these steps:

1. Identify the Denominators
   Collect all the denominators of the polynomial fractions you are working with.

2. Factor Each Denominator
   Factor each polynomial denominator into its irreducible components to identify the unique factors. This may involve factoring quadratic expressions, common factors, and looking for roots if applicable.

3. Determine the Least Common Denominator (LCD)
   Take the highest power of each unique factor that appears in any of the denominators. This means:

   • If a factor appears in a denominator multiple times, use the highest power of that factor.

   • If a factor is common across multiple denominators, select it once at its highest exponent.

4. Combine the Factors
   Multiply all selected factors together to obtain the least common denominator (LCD). This ensures that every denominator is a factor of the resulting common denominator.

When determining the Least Common Denominator (LCD) for polynomial fractions, using the highest power means you should look at each unique factor in the denominators and consider the largest exponent with which that factor appears across all denominators. For example:

• If you have the denominators:

  • $x^2$

  • $x^3$

  • $x^4$

• The factor $x$ appears with exponents 2, 3, and 4. To find the LCD, you would take $x$ to the highest power, which is $x^4$.

This ensures that the LCD can be divided by all the individual denominators without leaving a remainder, making it suitable for adding or subtracting the fractions.