Anybody know how to do this? I need help ASAP..Express (x^2-x)/((x^2+3)(x^2+2)) in partial fraction. Thanks a lot.
To express the rational function `(x^2-x)/((x^2+3)(x^2+2))` in partial fraction, follow these steps:
Step 1: Factor the denominator.
The denominator of the function, `(x^2+3)(x^2+2)`, can be factored as follows: `(x+i√3)(x-i√3)(x+√2)(x-√2)`.
Step 2: Set up the partial fraction decomposition.
The general form of the partial fraction decomposition is `(A/(x+i√3)) + (B/(x-i√3)) + (C/(x+√2)) + (D/(x-√2))`.
Step 3: Find the values of A, B, C, and D.
To find the values of A, B, C, and D, multiply the entire equation by the original denominator, `(x^2+3)(x^2+2)`, to remove the denominators. Then, equate the numerator of the resulting equation to `(x^2-x)`.
Step 4: Solve for A, B, C, and D.
To solve for A, B, C, and D, you can substitute specific values of x that make some of the terms in the partial fraction decomposition zero. For example, substituting x = -i√3 will make the term `(A/(x+i√3))` zero, allowing you to solve for the corresponding constant.
Repeat this process for each term in the partial fraction decomposition to find the values of A, B, C, and D.
Step 5: Express the rational function in partial fraction form.
Once you have found the values of A, B, C, and D, substitute them back into the partial fraction decomposition to express the original function in partial fraction form.
I hope this explanation helps you to understand the process of expressing the given rational function in partial fraction.