Method of Partial Fraction Decomposition

Oct 13, 2025 by leolavanderia.com.br 38 views

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Introduction

The given problem is to resolve the expression ${$ \frac{1}{x^4(x+1)}$}$. This involves simplifying the expression to its most basic form. To do this, we will use the method of partial fraction decomposition, which is a technique used to break down a rational function into simpler fractions.

Understanding the Problem

The given expression is a rational function, which is a ratio of two polynomials. In this case, the numerator is 1 and the denominator is ${$ x^4(x+1)$}$. Our goal is to simplify this expression by breaking it down into simpler fractions.

Method of Partial Fraction Decomposition

The method of partial fraction decomposition involves expressing a rational function as a sum of simpler fractions. This is done by assuming that the rational function can be expressed as a sum of simpler fractions, and then using algebraic techniques to determine the coefficients of these fractions.

Step 1: Factor the Denominator

The first step in using the method of partial fraction decomposition is to factor the denominator of the rational function. In this case, the denominator is ${$ x^4(x+1)$}$, which can be factored as ${$ x^4(x+1) = x^4(x+1)$}$.

Step 2: Write the Partial Fraction Decomposition

Once the denominator has been factored, we can write the partial fraction decomposition of the rational function. In this case, we can write:

${$ \frac{1}{x^4(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{E}{x+1}$}$

where A, B, C, D, and E are constants to be determined.

Step 3: Clear the Fractions

To determine the constants A, B, C, D, and E, we need to clear the fractions in the partial fraction decomposition. This is done by multiplying both sides of the equation by the least common multiple of the denominators, which is ${$ x^4(x+1)$}$.

Step 4: Expand and Simplify

Once the fractions have been cleared, we can expand and simplify the equation to determine the values of the constants A, B, C, D, and E.

Step 5: Solve for the Constants

To solve for the constants A, B, C, D, and E, we need to equate the coefficients of the terms on both sides of the equation. This will give us a system of equations that we can solve to determine the values of the constants.

Step 6: Write the Final Answer

Once we have determined the values of the constants A, B, C, D, and E, we can write the final answer as a sum of simpler fractions.

Solution

Let's go through the steps to solve the problem.

Step 1: Factor the Denominator

The denominator is already factored as ${$ x^4(x+1)$}$.

Step 2: Write the Partial Fraction Decomposition

We can write:

${$ \frac{1}{x^4(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} + \frac{E}{x+1}$}$

Step 3: Clear the Fractions

Multiplying both sides by ${$ x^4(x+1)$}$, we get:

${ $1 = A(x^3)(x+1) + B(x^2)(x+1) + C(x)(x+1) + D(x^4) + E(x^4)\$}$

Step 4: Expand and Simplify

Expanding and simplifying the equation, we get:

${ $1 = Ax^4 + Ax^3 + Bx^3 + Bx^2 + Cx^2 + Cx + Dx^4 + Ex^4\$}$

Step 5: Solve for the Constants

Equating the coefficients of the terms on both sides of the equation, we get:

${$ A + D = 0$}{ $A + B + D + E = 0\$}$ ${$ B + C + E = 0$}{ $C = 0\$}$

Step 6: Write the Final Answer

Solving the system of equations, we get:

${$ A = -\frac{1}{x^4}$}{ $B = \frac{1}{x^3}\$}$ ${$ C = 0$}{ $D = \frac{1}{x^4}\$}$ ${$ E = -\frac{1}{x^3}$}$

Therefore, the final answer is:

${$ \frac{1}{x^4(x+1)} = -\frac{1}{x^4} + \frac{1}{x^3} + \frac{1}{x^3} - \frac{1}{x^4}$}$

Simplifying the expression, we get:

${$ \frac{1}{x^4(x+1)} = \frac{1}{x^3} - \frac{1}{x^4}$}$

Conclusion

In this article, we have shown how to resolve the expression ${$ \frac{1}{x^4(x+1)}$}$ using the method of partial fraction decomposition. We have broken down the expression into simpler fractions and solved for the constants A, B, C, D, and E. The final answer is ${$ \frac{1}{x^3} - \frac{1}{x^4}$}$.

Final Answer

The final answer is ${$ \frac{1}{x^3} - \frac{1}{x^4}$}$.

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Introduction

In our previous article, we showed how to resolve the expression ${$ \frac{1}{x^4(x+1)}$}$ using the method of partial fraction decomposition. In this article, we will answer some common questions related to this topic.

Q: What is the method of partial fraction decomposition?

A: The method of partial fraction decomposition is a technique used to break down a rational function into simpler fractions. It involves expressing a rational function as a sum of simpler fractions, and then using algebraic techniques to determine the coefficients of these fractions.

Q: Why is the method of partial fraction decomposition useful?

A: The method of partial fraction decomposition is useful because it allows us to break down a complex rational function into simpler fractions, making it easier to analyze and solve. It is also useful in solving systems of equations and in calculus.

Q: How do I know when to use the method of partial fraction decomposition?

A: You should use the method of partial fraction decomposition when you have a rational function that cannot be simplified further using basic algebraic techniques. It is also useful when you need to break down a complex rational function into simpler fractions.

Q: What are some common mistakes to avoid when using the method of partial fraction decomposition?

A: Some common mistakes to avoid when using the method of partial fraction decomposition include:

Not factoring the denominator correctly
Not clearing the fractions correctly
Not solving for the constants correctly
Not checking the solution for errors

Q: How do I check my solution for errors?

A: To check your solution for errors, you should:

Plug in the values of the constants into the original equation
Simplify the equation to see if it matches the original equation
Check for any errors in the algebraic manipulations

Q: What are some common applications of the method of partial fraction decomposition?

A: Some common applications of the method of partial fraction decomposition include:

Solving systems of equations
Calculus
Differential equations
Electrical engineering
Signal processing

Q: Can the method of partial fraction decomposition be used to solve non-rational functions?

A: No, the method of partial fraction decomposition can only be used to solve rational functions. It is not applicable to non-rational functions.

Q: How do I know if a rational function can be simplified using the method of partial fraction decomposition?

A: You can determine if a rational function can be simplified using the method of partial fraction decomposition by factoring the denominator and checking if it can be broken down into simpler fractions.

Q: What are some common challenges when using the method of partial fraction decomposition?

A: Some common challenges when using the method of partial fraction decomposition include:

Factoring the denominator correctly
Clearing the fractions correctly
Solving for the constants correctly
Checking the solution for errors

Q: How do I overcome these challenges?

A: To overcome these challenges, you should:

Take your time and be careful when factoring the denominator
Double-check your work when clearing the fractions
Use algebraic techniques to solve for the constants
Check your solution carefully for errors

Conclusion

In this article, we have answered some common questions related to the method of partial fraction decomposition. We have discussed the importance of this technique, common mistakes to avoid, and common applications. We have also provided tips on how to overcome common challenges when using this technique.

Final Answer

The final answer is ${$ \frac{1}{x^3} - \frac{1}{x^4}$}$.