Class 12 Math|Ex 5.3|

Опубликовано: 18 Июнь 2026
на канале: Misha's Math HUB

Partial Fraction Decomposition is a method used in algebra to simplify rational functions. When you have a rational function that can be expressed as a fraction of two polynomials, you can use partial fraction decomposition to break it down into simpler fractions.

Here's a general overview of the steps involved in the partial fraction method:

1. **Factorize the denominator**: Start by factoring the denominator polynomial into its irreducible factors. This step involves finding the roots of the denominator polynomial.

2. **Write the partial fraction form**: Express the original rational function as a sum of simpler fractions with unknown constants as their numerators and the irreducible factors of the denominator as their denominators. For example, if the denominator has linear and quadratic factors, you would write the partial fraction form with terms like A/(x - a) + B/(x - b) + C/(x^2 + px + q).

3. **Find the constants**: Multiply both sides of the equation by the original denominator to clear the fractions. This will help you solve for the unknown constants by comparing the coefficients of like terms on both sides of the equation.

4. **Solve for the unknowns**: Once you have set up the equation with the partial fraction form, solve for the unknown constants using algebraic methods like equating coefficients or substitution.

5. **Check your solution**: After finding the values of the unknown constants, substitute them back into the partial fraction form and simplify. Ensure that your simplified expression matches the original rational function.

This method is particularly useful when integrating rational functions, as it can simplify complex integrals into more manageable parts.