Partial Fraction Decomposition Calculator

Numerator (e.g., x + 7):

Denominator (e.g., (x + 1)(x + 2)):

Partial fraction decomposition is a method used to decompose a rational function into simpler fractions. It's particularly useful in integration and solving linear differential equations. Here's a brief overview of how it works:

Steps for Partial Fraction Decomposition:

1. Factor the Denominator: Factor the denominator of the rational function if it is not already factored.

2. Write the Decomposition: Express the rational function as a sum of simpler fractions where each denominator is a factor of the original denominator.

   For example, if the denominator is factored into linear and quadratic factors, the decomposition might look like:

   $$\frac{P(x)}{Q(x)} = \frac{A_1}{(x - r_1)} + \frac{A_2}{(x - r_2)} + \ldots + \frac{A_n}{(x^2 + bx + c)}$$

   Here, $P(x)$ and $Q(x)$ are polynomials, and $A_1, A_2, \ldots, A_n$ are constants to be determined.

3. Determine Constants: To find the constants $A_1, A_2, \ldots, A_n$:

   • Multiply both sides by the denominator of each individual fraction to eliminate the denominators.
   • Substitute appropriate values of $x$ to solve for each constant.

4. Set up Equations: After substituting values of $x$, you'll get a system of linear equations. Solve this system to find the values of $A_1, A_2, \ldots, A_n$.

5. Write the Final Answer: Once you have determined the values of the constants $A_1, A_2, \ldots, A_n$, substitute them back into the original partial fraction decomposition equation to get the final decomposition.

Example

Consider the function:

$\frac{4x + 5}{(x-1)(x+2)}$

1. Factor the Denominator: $(x-1)(x+2)$

2. Set Up Partial Fractions: $\frac{4x + 5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

3. Combine Fractions: $\frac{A(x+2) + B(x-1)}{(x-1)(x+2)} = \frac{4x + 5}{(x-1)(x+2)}$

4. Solve for the Unknowns: $A(x+2) + B(x-1) = 4x + 5$ Equate coefficients:

   • $A + B = 4$
   • $2A - B = 5$

   Solving these equations:

   • Add the equations: $3A = 9 \Rightarrow A = 3$
   • Substitute $A$ into $A + B = 4$: $3 + B = 4 \Rightarrow B = 1$

5. Rewrite the Original Fraction: $\frac{4x + 5}{(x-1)(x+2)} = \frac{3}{x-1} + \frac{1}{x+2}$

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