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 a partial fraction decomposition for a rational function with distinct linear factors?

Back

The general form is: \( \frac{A_1}{(x - r_1)} + \frac{A_2}{(x - r_2)} + \ldots + \frac{A_n}{(x - r_n)} \) where \( r_1, r_2, \ldots, r_n \) are the roots of the denominator.

Front

How do you determine the coefficients in a partial fraction decomposition?

Back

You multiply both sides of the equation by the common denominator and then equate coefficients for corresponding powers of x.

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 variable 'B' in partial fraction decomposition?

Back

'B' represents the coefficient of the term associated with a specific factor in the denominator, which needs to be determined during the decomposition process.

Front

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

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'A' is the coefficient for the first term in the decomposition, representing the contribution of that factor to the overall function.

Front

How do you handle repeated linear factors in partial fraction decomposition?

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For repeated linear factors, the decomposition includes terms for each power of the factor, such as \( \frac{A_1}{(x - r)} + \frac{A_2}{(x - r)^2} + \ldots + \frac{A_n}{(x - r)^n} \).

Front

What is the purpose of substituting specific values for x when solving for coefficients in partial fraction decomposition?

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Substituting specific values for x simplifies the equation, allowing you to solve for the unknown coefficients directly.

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 partial fraction decomposition of \( \frac{x^2-8}{x^2(x-1)} \)?

Back

The partial fraction decomposition is \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{(x-1)} \).