Decompose rational functions into partial fractions

Understanding Rational Functions and Partial Fractions

A rational function is any function that can be defined by a ratio of two polynomials. In other words, it is a function in the form (R(x) = P(x)/Q(x)), where (P(x)) and (Q(x)) are polynomials.
Partial fractions allow us to separate a complex rational function into a sum of simpler fractions, making it easier to integrate, differentiate, and simplify. This process is called decomposition.

Prerequisites for Decomposition

Before decomposing a rational function into partial fractions, you must first ensure that the degree of the numerator polynomial is less than the degree of the denominator polynomial. If it’s not, you should use polynomial division to rewrite the function, so it is.
Additionally, you must fully factorise the denominator polynomial.

Decomposing Rational Functions

If the denominator factors into linear factors (i.e., they are to the power of 1), write it as a sum of fractions where each denominator is one of these linear factors. The numerators are unknown coefficients, which you will need to solve for.
If the denominator has repeated linear factors (i.e., the same factor occurs more than once), you sum over all possible powers.
If there are irreducible quadratic factors in the denominator, you will have a numerator of the form of (ax + b), where a and b are constants.

Determining Coefficients

To find the values of the unknown coefficients in the numerator of each of the partial fractions, substitute convenient values of x into the equation then solve for the coefficients.
In some cases, it may be necessary to differentiate or use simultaneous equations.

Common Uses of Partial Fractions in Further Mathematics

Decomposing rational functions into partial fractions is particularly useful for performing operations such as integration and differentiation, where the simplicity of the partial fractions can greatly simplify the procedure.
They’re also used in solving linear ordinary differential equations.
Moreover, knowing how to decompose into partial fractions is key in complex number theory and in understanding residues and contours within the study of complex analysis.

Remember, practice and familiarity with different rational functions and their corresponding decompositions into partial fractions is essential. This is a backbone procedure in further mathematics and mastering it will enable you tackle many different styles of problem.