Jbreezy said:Homework Statement
Write out the form of the partial fraction decomposition of not determine the numerical values of the coefficients.
Homework Equations
x^4 -2x^3 + x^2 +2x -1 / x^2 -2x +1
The Attempt at a Solution
I did the division and I got x^2 + ((2x-1)) / (x^2 -2x +1)
So I took
((2x-1)) / (x^2 -2x +1) = Ax + B / x^2 -2x +1
Is this right? It has been forever since I did these.
Partial fractions are a method of breaking down a rational expression into simpler fractions. This can make it easier to solve equations involving fractions or integrate functions.
Partial fractions are typically used when dealing with rational expressions that involve polynomials of different degrees in the numerator and denominator. They can also be used when integrating rational functions.
To find the partial fraction decomposition, you first factor the denominator of the rational expression. Then, you set up an equation with unknown coefficients for each factor. From there, you solve for the coefficients by equating the original rational expression to the partial fraction decomposition.
Sure! Let's say we have the equation (x+1)/(x^2+3x+2). We first factor the denominator as (x+2)(x+1). Then, we set up the equation A/(x+2) + B/(x+1) = (x+1)/(x^2+3x+2). Solving for A and B, we get A = 1 and B = -1. Therefore, the partial fraction decomposition is 1/(x+2) - 1/(x+1).
Yes, there are a few special cases to keep in mind when using partial fractions. These include when the denominator has repeated factors, when the degree of the numerator is greater than or equal to the degree of the denominator, and when the denominator has complex roots. In these cases, additional steps may be necessary to find the partial fraction decomposition.