Partial Fraction Decomposition

Front

What is partial fraction decomposition?

Back

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

Front

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

Back

The general form includes terms for each power of the factor, such as \( \frac{A}{(x-a)} + \frac{B}{(x-a)^2} + \ldots \) for a repeated factor \( (x-a)^n \).

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 \( \frac{A}{x} \) in partial fraction decomposition?

Back

It represents the contribution of the simple linear factor \( x \) in the denominator, where \( A \) is a constant to be determined.

Front

What is the significance of the term \( \frac{B}{x^2} \) in partial fraction decomposition?

Back

It accounts for the repeated linear factor \( x^2 \) in the denominator, where \( B \) is a constant to be determined.

Front

What is the significance of the term \( \frac{C}{(x-1)} \) in partial fraction decomposition?

Back

It represents the contribution of the linear factor \( (x-1) \) in the denominator, where \( C \) is a constant to be determined.

Front

How do you handle irreducible quadratic factors in partial fraction decomposition?

Back

For irreducible quadratic factors, use the form \( \frac{Ax + B}{(x^2 + bx + c)} \) where A and B are constants to be determined.

Front

What is the purpose of multiplying the entire equation by the common denominator in partial fraction decomposition?

Back

Multiplying by the common denominator eliminates the fractions, allowing for easier manipulation and comparison of coefficients.

Front

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

Back

'B' is a constant that represents the coefficient of the term associated with a repeated linear factor in the decomposition.