# Partial Fractions

**Partial Fractions**

**Understanding Partial Fractions**

**Partial fractions**involve the decomposition of a complex fraction into simpler fractions. This technique is useful in calculus and differential equations.- The
**numerator**must be of a lower order (degree) than the denominator for a fraction to be split into partial fractions. - It is necessary to perform
**long division**before partial fractions can be used if this condition is not met.

**Types of Partial Fractions**

**Proper Fractions**: The degree of the numerator is less than the degree of the denominator.**Improper Fractions**: The degree of the numerator is greater than or equal to the degree of the denominator.**Linear Partial Fractions**: The factors of the denominator are linear (i.e., they are to the power of 1).**Repeated Linear Partial Fractions**: The factors of the denominator are the same linear factor repeated.**Irreducible Quadratic Factors**: The denominator can have factors that are irreducible quadratics.

**Techniques to Solve Partial Fractions**

**Factorise the denominator**: The first step in splitting fractions into partial fractions is to factorise the denominator.**Split into partial fractions**: After factorising the denominator, express the fraction as the sum of simpler fractions with unknown numerators.**Finding the unknowns**: Equate coefficients or substitute suitable values to find the values of the unknowns.

**Benefits of using Partial Fractions**

**Simplify integration and differentiation**: Calculating the integral or derivative of multiple simple fractions can be easier than calculating the integral or derivative of a single complex fraction.**Simplify algebraic manipulation**: Partial fractions can make solving equations or rearranging expressions easier.

**Common mistakes**

**Not fully factorising the denominator**: Ensure that the denominator is fully factorised into linear or irreducible quadratic factors to ensure the correct form of the partial fractions.**Overlooking repeated roots / factors**: If a factor of the denominator is repeated, a partial fraction is needed for each occurrence of the factor.**Incorrect degree of the numerator**: Make sure the degree of the numerator in each partial fraction is one less than the degree of the factor in the denominator.