# Partial Fraction: Solving x^3+1/x^2+4

• tnutty
In summary, the problem involves finding the partial fraction decomposition of the expression x^3+1/x^2+4 after using long division. The first part of the decomposition is straightforward, but the second part may require a trig substitution to solve.
tnutty

x^3+1 / x^2+4

## The Attempt at a Solution

after using long division i get : x + -(4x-1) / (x^2+4)

then using partial fraction :

-4x-1 / x^2+4 = bx+c / x^2+4

then i am stuck because if what i have
is correct then they both are the same and in the same form.

tnutty said:
-4x-1 / x^2+4 = bx+c / x^2+4

then i am stuck because if what i have
is correct then they both are the same and in the same form.

$$-\frac{4x-1}{x^2+4}=\frac{1}{x^2+4}-\frac{4x}{x^2+4}$$...

Once you gotten to (-4x-1)/(x^2+4) you probably want to leave the partial fractions trail. Split it into -4x/(x^2+4) and -1/(x^2+4). The first one is a simple substitution. I'm going to guess the second one might be something related to arctan(x)? Try a trig substitution.

## 1. What is a partial fraction?

A partial fraction is an algebraic expression that can be divided into simpler fractions. It is used to solve complex equations, such as x^3+1/x^2+4, by breaking it down into smaller parts that are easier to solve.

## 2. How do you solve x^3+1/x^2+4 using partial fractions?

To solve x^3+1/x^2+4 using partial fractions, you need to first factor the denominator, which in this case is x^2+4. Then, you need to find the coefficients for each factor by setting up a system of equations and solving for them. Once you have the coefficients, you can rewrite the equation as a sum of simpler fractions and solve each one individually.

## 3. Can all equations be solved using partial fractions?

No, not all equations can be solved using partial fractions. This method is usually used for rational functions that can be factored into simpler fractions. If an equation cannot be factored, then it cannot be solved using partial fractions.

## 4. What are the benefits of using partial fractions to solve equations?

One benefit of using partial fractions is that it allows you to break down a complex equation into simpler parts, making it easier to solve. It also helps to find the general solution to an equation, as it allows you to find the coefficients for each factor.

## 5. Are there any drawbacks to using partial fractions?

One drawback of using partial fractions is that the process can be time-consuming and tedious. It also requires a good understanding of algebra and factoring to correctly solve the equations. Additionally, not all equations can be solved using this method, so it may not always be applicable.

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