Understanding Partial Fractions Decomposition

Oct 13, 2025 by pynkocean.com 125 views

Introduction

Partial fractions decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable components. This method involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. In this article, we will explore the concept of partial fractions decomposition and apply it to the given problem: splitting $\frac{4x+5}{x^2-2x+1}$ into partial fractions.

Understanding Partial Fractions Decomposition

Partial fractions decomposition is based on the concept of expressing a rational function as a sum of simpler fractions. The general form of a rational function is:

\frac{f(x)}{g(x)}

where $f(x)$ and $g(x)$ are polynomials. To decompose this rational function into partial fractions, we need to express it as a sum of simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.

The Process of Partial Fractions Decomposition

The process of partial fractions decomposition involves the following steps:

Factor the denominator: The first step is to factor the denominator of the rational function. In this case, the denominator is $x^2-2x+1$ , which can be factored as $(x-1)^2$ .
Express the rational function as a sum of simpler fractions: Once the denominator is factored, we can express the rational function as a sum of simpler fractions. The general form of a partial fraction is:

\frac{A}{x-a} + \frac{B}{(x-a)^2}

where $A$ and $B$ are constants to be determined. 3. Clear the fractions: To clear the fractions, we multiply both sides of the equation by the common denominator, which is $(x-1)^2$ in this case. 4. Solve for the constants: Once the fractions are cleared, we can solve for the constants $A$ and $B$ by equating the coefficients of the polynomials.

Applying Partial Fractions Decomposition to the Given Problem

Now that we have a good understanding of the process of partial fractions decomposition, let's apply it to the given problem: splitting $\frac{4x+5}{x^2-2x+1}$ into partial fractions.

Step 1: Factor the denominator

The denominator of the rational function is $x^2-2x+1$ , which can be factored as $(x-1)^2$ .

Step 2: Express the rational function as a sum of simpler fractions

We can express the rational function as a sum of simpler fractions:

\frac{4x+5}{(x-1)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2}

Step 3: Clear the fractions

To clear the fractions, we multiply both sides of the equation by the common denominator, which is $(x-1)^2$ in this case:

4x+5 = A(x-1) + B

Step 4: Solve for the constants

Now that the fractions are cleared, we can solve for the constants $A$ and $B$ by equating the coefficients of the polynomials. We can do this by substituting values of $x$ that will make the equation easier to solve.

Let's substitute $x=1$ into the equation:

4(1)+5 = A(1-1) + B

Simplifying the equation, we get:

9 = B

Now that we have found the value of $B$ , we can substitute it back into the original equation:

4x+5 = A(x-1) + 9

We can now solve for the value of $A$ by equating the coefficients of the polynomials. We can do this by substituting values of $x$ that will make the equation easier to solve.

Let's substitute $x=0$ into the equation:

5 = A(-1) + 9

Simplifying the equation, we get:

-4 = A

Now that we have found the values of $A$ and $B$ , we can express the rational function as a sum of simpler fractions:

\frac{4x+5}{(x-1)^2} = \frac{-4}{x-1} + \frac{9}{(x-1)^2}

Conclusion

In this article, we have applied the technique of partial fractions decomposition to the given problem: splitting $\frac{4x+5}{x^2-2x+1}$ into partial fractions. We have factored the denominator, expressed the rational function as a sum of simpler fractions, cleared the fractions, and solved for the constants. The final answer is:

\frac{4x+5}{x^2-2x+1} = \frac{-4}{x-1} + \frac{9}{(x-1)^2}

This is the correct answer, which can be verified by multiplying the partial fractions back together.

Final Answer

The final answer is:

Introduction

Partial fractions decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable components. In our previous article, we explored the concept of partial fractions decomposition and applied it to the given problem: splitting $\frac{4x+5}{x^2-2x+1}$ into partial fractions. In this article, we will answer some frequently asked questions about partial fractions decomposition.

Q: What is partial fractions decomposition?

A: Partial fractions decomposition is a technique used to simplify complex rational expressions into more manageable components. It involves expressing a rational function as a sum of simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.

Q: When can I use partial fractions decomposition?

A: You can use partial fractions decomposition when you have a rational function with a polynomial numerator and a linear or quadratic denominator. This technique is particularly useful when you need to integrate or differentiate a rational function.

Q: How do I factor the denominator?

A: To factor the denominator, you need to find the roots of the polynomial. You can do this by using the quadratic formula or by factoring the polynomial by grouping.

Q: What is the general form of a partial fraction?

A: The general form of a partial fraction is:

\frac{A}{x-a} + \frac{B}{(x-a)^2}

where $A$ and $B$ are constants to be determined.

Q: How do I clear the fractions?

A: To clear the fractions, you multiply both sides of the equation by the common denominator.

Q: How do I solve for the constants?

A: To solve for the constants, you need to equate the coefficients of the polynomials. You can do this by substituting values of $x$ that will make the equation easier to solve.

Q: What are some common mistakes to avoid when using partial fractions decomposition?

A: Some common mistakes to avoid when using partial fractions decomposition include:

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

Q: How do I check my final answer?

A: To check your final answer, you can multiply the partial fractions back together and simplify the expression. If the final answer is correct, the expression should be equal to the original rational function.

Q: What are some real-world applications of partial fractions decomposition?

A: Partial fractions decomposition has many real-world applications, including:

Integrating and differentiating rational functions
Solving systems of linear equations
Finding the area under curves
Modeling population growth and decay

Conclusion

In this article, we have answered some frequently asked questions about partial fractions decomposition. We have covered topics such as the definition of partial fractions decomposition, when to use it, how to factor the denominator, and how to solve for the constants. We have also discussed some common mistakes to avoid and how to check your final answer. By following these guidelines, you can master the technique of partial fractions decomposition and apply it to a wide range of problems.

Final Answer

The final answer is:

$\boxed{\frac{-4}{x-1} + \frac{9}{(x-1)^2}}$