Partial Fractions
Partial fractions are a technique used to decompose rational functions into simpler components. This method is particularly useful in calculus, especially for integration, and in solving certain types of differential equations, discussed further in topic 5.
Decomposition of Rational Functions
The process of partial fraction decomposition involves breaking down a complex rational function into a sum of simpler fractions. To perform partial fraction decomposition, the following conditions need to be met:
- The polynomial degree of the numerator is less than the degree of the denominator (e.g. \frac{x-4}{x^2+4x+3}
- The denominator can be factored into a product of linear terms (at most two distinct linear terms)
The condition that the numerator's degree must be less than the denominator's is crucial. If this is not the case, long division must be performed first to obtain a polynomial remainder and a rational function that meets this condition.
General Form of Partial Fractions
For a rational function with two distinct linear factors in the denominator, the general form of partial fraction decomposition is:
$$\frac{P(x)}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}$$
Where $P(x)$ is a polynomial of degree less than 2, and $A$ and $B$ are constants to be determined.
Method for Finding Coefficients
To find the coefficients $A$ and $B$, we can use the following steps:
- Multiply both sides of the equation by the denominator of the original fraction.
- Expand and simplify the right-hand side.
- Equate coefficients of like terms on both sides.
- Solve the resulting system of linear equations.
Let's decompose $\frac{2x+1}{x^2+x-2}$ into partial fractions.
Step 1: Factor the denominator: $x^2+x-2 = (x-1)(x+2)$
Step 2: Set up the partial fraction form:
$$\frac{2x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$
Step 3: Multiply both sides by $(x-1)(x+2)$:
$2x+1 = A(x+2) + B(x-1)$
Step 4: Expand the right-hand side:
$2x+1 = Ax + 2A + Bx - B = (A+B)x + (2A-B)$
Step 5: Equate coefficients:
$2 = A+B$ $1 = 2A-B$
Step 6: Solve the system of equations:
Adding the equations: $3 = 3A$, so $A = 1$ Substituting back: $B = 2-A = 1$
Therefore, the partial fraction decomposition is:
$$\frac{2x+1}{x^2+x-2} = \frac{1}{x-1} + \frac{1}{x+2}$$
