Why Decompose?

Partial fraction decomposition rewrites a rational function as the sum of simpler fractions. For example, $\frac{ax+b}{(x-\mathrm{r\_1})(x-\mathrm{r\_2})}$ becomes $\frac{A}{x-\mathrm{r\_1}}+\frac{B}{x-\mathrm{r\_2}}$. This technique simplifies integration and highlights the poles of the function, which is useful in engineering and differential equations.

Setting Up the Problem

To decompose, we equate coefficients. Multiplying both sides of the equation above by the denominator yields $ax+b=A(x-\mathrm{r\_2})+B(x-\mathrm{r\_1})$. Expanding and collecting like terms provides two linear equations in the unknowns $A$ and $B$. Solving these gives the partial fraction coefficients.

Using the Calculator

Enter the numerator coefficients $a$ and $b$, along with the roots $\mathrm{r\_1}$ and $\mathrm{r\_2}$ of the denominator. The calculator solves for $A$ and $B$ by substituting into the equations derived above. Results appear in the form $\frac{A}{x-\mathrm{r\_1}}+\frac{B}{x-\mathrm{r\_2}}$, which you can use directly for integration or inverse Laplace transforms.

A Worked Example

Suppose $a=3$, $b=5$, $\mathrm{r\_1}=1$, and $\mathrm{r\_2}=2$. The system of equations becomes $3x+5=A(x-2)+B(x-1)$. Expanding yields $(A+B)x+(-2A-B)$. Equating coefficients leads to $A+B=3$ and $-2A-B=5$. Solving gives $A=-8$ and $B=11$.

Applications

Partial fraction decomposition simplifies integrals involving rational expressions. In control theory and circuits, it helps invert Laplace transforms, turning complex transfer functions into sums of easier terms. In algebra, it assists with partial fraction expansions over the real or complex numbers, revealing poles and residues. Understanding this technique builds intuition about the structure of rational functions.

Historical Background

The method dates back to the 1700s, with contributions from mathematicians such as Heaviside, who used it to simplify differential operators. Its continued relevance shows how algebraic techniques evolve yet remain essential in modern analysis. By mastering partial fractions, you gain a versatile tool for calculus, engineering, and beyond.

Tips for Success

Check that the denominator factors into distinct linear terms. If the roots coincide, you need repeated fractions with higher powers in the denominator. This calculator focuses on distinct real roots for simplicity. Always verify your results by combining the partial fractions and confirming that the original numerator reappears. The more examples you try, the more comfortable you will become with the process.

Explore how the coefficients change when you vary the roots or the numerator. Notice how poles near each other produce larger coefficients, reflecting their influence on the function's shape. This insight helps when approximating systems or evaluating integrals in applied contexts.