Partial Fraction Decomposition

Front

What is partial fraction decomposition?

Back

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions, making it easier to integrate or manipulate.

Front

What is the general form of partial fraction decomposition for a rational function with repeated linear factors?

Back

The general form is: \( \frac{A}{(x-r)} + \frac{B}{(x-r)^2} + \ldots + \frac{C}{(x-s)} \) where \( r \) and \( s \) are the roots of the denominator.

Front

How do you determine the coefficients in partial fraction decomposition?

Back

You multiply both sides of the equation by the common denominator, then equate coefficients for corresponding powers of x to solve for the unknowns.

Front

What is the first step in performing partial fraction decomposition?

Back

The first step is to ensure that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first.

Front

What is the significance of the term \( (x+3)^2 \) in the denominator of \( \frac{x-11}{(x+3)^2(x-4)} \)?

Back

It indicates that the partial fraction decomposition will include a term for both \( \frac{A}{(x+3)} \) and \( \frac{B}{(x+3)^2} \) due to the repeated factor.

Front

What is the partial fraction decomposition of \( \frac{10x-11}{(x+1)(2x-5)} \)?

Back

The partial fraction decomposition is \( \frac{3}{(x+1)} + \frac{4}{(2x-5)} \).

Front

What is the purpose of setting up partial fractions?

Back

Setting up partial fractions simplifies the integration process by breaking down complex rational functions into simpler components.

Front

What do you do if the denominator has irreducible quadratic factors?

Back

For irreducible quadratic factors, include terms of the form \( \frac{Ax + B}{(ax^2 + bx + c)} \) in the decomposition.

Front

How do you handle a denominator with both linear and quadratic factors?

Back

Include separate terms for each linear factor and a term for each quadratic factor in the decomposition.

Front

What is the role of the variable 'B' in the context of partial fraction decomposition?

Back

'B' represents the coefficient of the term associated with the repeated linear factor in the decomposition.