The formula to calculate the fractional decomposition of a rational function is:
\[ \frac{A}{B} = \frac{A1}{B1} + \frac{A2}{B2} + \cdots + \frac{An}{Bn} \]
Where:
Fractional decomposition is a mathematical process used to break down complex fractions or rational expressions into simpler parts, often for the purpose of integration or simplification. It involves expressing the fraction as a sum of simpler fractions with linear or quadratic denominators. This method is particularly useful in calculus and algebra for solving equations and integrating functions.
Let's assume the following rational function:
\[ \frac{2x + 3}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2} \]
To find \(A\) and \(B\), we multiply both sides by \((x - 1)(x + 2)\):
\[ 2x + 3 = A(x + 2) + B(x - 1) \]
Next, we solve for \(A\) and \(B\) by setting up equations for the coefficients:
For \(x\): \(2 = A + B\)
For the constant term: \(3 = 2A - B\)
Solving these equations gives us \(A = 1\) and \(B = 1\).
Thus, the fractional decomposition is:
\[ \frac{2x + 3}{(x - 1)(x + 2)} = \frac{1}{x - 1} + \frac{1}{x + 2} \]
