Istruzioni passo passo
Factor the Denominator
First, factor the denominator into its simplest form. This will help you identify the roots of the denominator. For example, if the denominator is \( x^2 + 5x + 6 \), factor it as \( (x + 3)(x + 2) \).
Write the Partial Fraction Decomposition Form
Write the partial fraction decomposition form with the factored denominator. Using the example from Step 1, the form would be: \[ rac{N(x)}{(x + 3)(x + 2)} = rac{A}{x + 3} + rac{B}{x + 2} \].
Clear the Fractions
Multiply both sides of the equation by the factored denominator to clear the fractions. This will give you an equation with the numerator on one side and the partial fraction decomposition on the other. For example: \[ N(x) = A(x + 2) + B(x + 3) \].
Find the Constants
Choose suitable values for \( x \) to find the constants \( A \) and \( B \). For example, if \( x = -3 \), the equation becomes \( N(-3) = A(-3 + 2) \), which simplifies to \( N(-3) = -A \). Solving for \( A \) gives you the value of the constant. Repeat this process for \( B \) by choosing \( x = -2 \).
Combine the Partial Fractions
Once you have found the values of the constants, combine the partial fractions to get the final decomposition. Using the example, if \( A = 2 \) and \( B = 3 \), the decomposition would be: \[ rac{N(x)}{(x + 3)(x + 2)} = rac{2}{x + 3} + rac{3}{x + 2} \].
Check Your Work
Finally, check your work by plugging the decomposition back into the original equation to ensure it is true. You can also use a calculator to verify your result for convenience.
Introduction to Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is essential in various mathematical and scientific applications, such as calculus, algebra, and engineering.
Formula
The partial fraction decomposition formula is: [ rac{N(x)}{D(x)} = rac{A}{x - r_1} + rac{B}{x - r_2} + \cdots + rac{K}{x - r_n} ] where ( N(x) ) is the numerator, ( D(x) ) is the denominator, ( r_i ) are the roots of the denominator, and ( A, B, \ldots, K ) are constants.