Solve The Partial Fraction For:${ \frac{x 3+4x 2+20x-7}{(x-1) 2(x 2+8)} }$

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Introduction

Partial fractions are a powerful tool in algebra, used to break down complex rational functions into simpler components. In this article, we will focus on solving partial fractions for the given rational function:

x3+4x2+20xβˆ’7(xβˆ’1)2(x2+8)\frac{x^3+4x^2+20x-7}{(x-1)^2(x^2+8)}

We will walk through the step-by-step process of solving partial fractions, using the given rational function as an example.

What are Partial Fractions?

Partial fractions are a way to express a rational function as a sum of simpler fractions. This is useful when we need to integrate or differentiate a rational function, or when we need to solve equations involving rational functions.

A rational function is a function that can be written in the form:

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

where f(x)f(x) and g(x)g(x) are polynomials.

The Method of Partial Fractions

The method of partial fractions involves breaking down the rational function into simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.

The general form of a partial fraction decomposition is:

f(x)g(x)=A(xβˆ’a)n+B(xβˆ’b)m+β‹―\frac{f(x)}{g(x)} = \frac{A}{(x-a)^n} + \frac{B}{(x-b)^m} + \cdots

where AA, BB, etc. are constants, and aa, bb, etc. are the roots of the denominator.

Step 1: Factor the Denominator

The first step in solving partial fractions is to factor the denominator of the rational function.

In this case, the denominator is:

(xβˆ’1)2(x2+8)(x-1)^2(x^2+8)

We can factor the quadratic term x2+8x^2+8 as:

(x2+8)=(x+2i)(xβˆ’2i)(x^2+8) = (x+2i)(x-2i)

where ii is the imaginary unit.

So, the factored form of the denominator is:

(xβˆ’1)2(x+2i)(xβˆ’2i)(x-1)^2(x+2i)(x-2i)

Step 2: Write the Partial Fraction Decomposition

Now that we have factored the denominator, we can write the partial fraction decomposition.

The general form of the partial fraction decomposition is:

x3+4x2+20xβˆ’7(xβˆ’1)2(x+2i)(xβˆ’2i)=A(xβˆ’1)2+Bx+2i+Cxβˆ’2i\frac{x^3+4x^2+20x-7}{(x-1)^2(x+2i)(x-2i)} = \frac{A}{(x-1)^2} + \frac{B}{x+2i} + \frac{C}{x-2i}

where AA, BB, and CC are constants.

Step 3: Clear the Fractions

To clear the fractions, we multiply both sides of the equation by the denominator:

(xβˆ’1)2(x+2i)(xβˆ’2i)(x3+4x2+20xβˆ’7(xβˆ’1)2(x+2i)(xβˆ’2i))=(xβˆ’1)2(x+2i)(xβˆ’2i)(A(xβˆ’1)2+Bx+2i+Cxβˆ’2i)(x-1)^2(x+2i)(x-2i) \left( \frac{x^3+4x^2+20x-7}{(x-1)^2(x+2i)(x-2i)} \right) = (x-1)^2(x+2i)(x-2i) \left( \frac{A}{(x-1)^2} + \frac{B}{x+2i} + \frac{C}{x-2i} \right)

This simplifies to:

x3+4x2+20xβˆ’7=A(x+2i)(xβˆ’2i)+B(xβˆ’1)2(xβˆ’2i)+C(xβˆ’1)2(x+2i)x^3+4x^2+20x-7 = A(x+2i)(x-2i) + B(x-1)^2(x-2i) + C(x-1)^2(x+2i)

Step 4: Solve for the Constants

Now we need to solve for the constants AA, BB, and CC.

We can do this by choosing values of xx that will make some of the terms on the right-hand side disappear.

For example, if we choose x=1x=1, the terms involving BB and CC will disappear, leaving us with:

13+4(1)2+20(1)βˆ’7=A(1+2i)(1βˆ’2i)1^3+4(1)^2+20(1)-7 = A(1+2i)(1-2i)

Simplifying, we get:

9=A(5)9 = A(5)

So, A=95A = \frac{9}{5}.

Step 5: Find the Other Constants

We can use a similar process to find the other constants BB and CC.

For example, if we choose x=βˆ’2ix=-2i, the terms involving AA and CC will disappear, leaving us with:

(βˆ’2i)3+4(βˆ’2i)2+20(βˆ’2i)βˆ’7=B(βˆ’2iβˆ’1)2(βˆ’2iβˆ’2i)(-2i)^3+4(-2i)^2+20(-2i)-7 = B(-2i-1)^2(-2i-2i)

Simplifying, we get:

βˆ’56iβˆ’28=B(βˆ’4i)2-56i-28 = B(-4i)^2

So, B=βˆ’56iβˆ’2816=βˆ’74iβˆ’74B = \frac{-56i-28}{16} = -\frac{7}{4}i-\frac{7}{4}.

Step 6: Write the Final Answer

Now that we have found all the constants, we can write the final answer:

x3+4x2+20xβˆ’7(xβˆ’1)2(x2+8)=95(xβˆ’1)2+βˆ’74iβˆ’74x+2i+95xβˆ’2i\frac{x^3+4x^2+20x-7}{(x-1)^2(x^2+8)} = \frac{\frac{9}{5}}{(x-1)^2} + \frac{-\frac{7}{4}i-\frac{7}{4}}{x+2i} + \frac{\frac{9}{5}}{x-2i}

Conclusion

Solving partial fractions involves breaking down a rational function into simpler fractions, each with a polynomial numerator and a linear or quadratic denominator. We used the method of partial fractions to solve the given rational function, and found the partial fraction decomposition.

The final answer is a sum of three simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.

Example Use Cases

Partial fractions are used in a variety of applications, including:

  • Integration: Partial fractions can be used to integrate rational functions.
  • Differentiation: Partial fractions can be used to differentiate rational functions.
  • Solving Equations: Partial fractions can be used to solve equations involving rational functions.

Tips and Tricks

  • Factor the Denominator: The first step in solving partial fractions is to factor the denominator.
  • Write the Partial Fraction Decomposition: The general form of the partial fraction decomposition is:

f(x)g(x)=A(xβˆ’a)n+B(xβˆ’b)m+β‹―\frac{f(x)}{g(x)} = \frac{A}{(x-a)^n} + \frac{B}{(x-b)^m} + \cdots

  • Clear the Fractions: To clear the fractions, multiply both sides of the equation by the denominator.
  • Solve for the Constants: Choose values of xx that will make some of the terms on the right-hand side disappear.

Introduction

In our previous article, we discussed the method of partial fractions and how to solve partial fractions for a given rational function. In this article, we will answer some frequently asked questions about partial fractions.

Q: What is the purpose of partial fractions?

A: The purpose of partial fractions is to break down a complex rational function into simpler components, making it easier to integrate, differentiate, or solve equations involving rational functions.

Q: How do I know when to use partial fractions?

A: You should use partial fractions when you have a rational function that cannot be easily integrated or differentiated, or when you need to solve an equation involving a rational function.

Q: What is the difference between partial fractions and polynomial division?

A: Polynomial division is a method of dividing a polynomial by another polynomial, resulting in a quotient and a remainder. Partial fractions, on the other hand, is a method of breaking down a rational function into simpler components.

Q: Can I use partial fractions to solve equations involving rational functions?

A: Yes, partial fractions can be used to solve equations involving rational functions. By breaking down the rational function into simpler components, you can solve the equation more easily.

Q: How do I choose the values of x to use in the partial fraction decomposition?

A: You should choose values of x that will make some of the terms on the right-hand side disappear. This will allow you to solve for the constants in the partial fraction decomposition.

Q: What if I have a rational function with a repeated root?

A: If you have a rational function with a repeated root, you will need to use a different method to solve the partial fraction decomposition. This method involves using the root to create a new polynomial that can be factored.

Q: Can I use partial fractions to solve equations involving complex rational functions?

A: Yes, partial fractions can be used to solve equations involving complex rational functions. However, you will need to use complex numbers and complex arithmetic to solve the equation.

Q: How do I know if I have found the correct partial fraction decomposition?

A: You can check if you have found the correct partial fraction decomposition by plugging the original rational function back into the equation and simplifying. If the equation is true, then you have found the correct partial fraction decomposition.

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

A: Some common mistakes to avoid when solving partial fractions include:

  • Not factoring the denominator correctly
  • Not choosing the correct values of x to use in the partial fraction decomposition
  • Not solving for the constants correctly
  • Not checking the equation to make sure it is true

Conclusion

Solving partial fractions can be a challenging task, but with practice and patience, you can master the method. By following the steps outlined in this article and avoiding common mistakes, you can solve partial fractions and apply them to a variety of applications.

Example Use Cases

Partial fractions are used in a variety of applications, including:

  • Integration: Partial fractions can be used to integrate rational functions.
  • Differentiation: Partial fractions can be used to differentiate rational functions.
  • Solving Equations: Partial fractions can be used to solve equations involving rational functions.
  • Signal Processing: Partial fractions can be used to analyze and design filters in signal processing.
  • Control Systems: Partial fractions can be used to analyze and design control systems.

Tips and Tricks

  • Practice, Practice, Practice: The more you practice solving partial fractions, the more comfortable you will become with the method.
  • Use a Calculator: A calculator can be a big help when solving partial fractions, especially when dealing with complex numbers and complex arithmetic.
  • Check Your Work: Always check your work to make sure the equation is true.
  • Use a Graphing Calculator: A graphing calculator can be a big help when visualizing the partial fraction decomposition and checking the equation.

By following these tips and tricks, you can master the method of partial fractions and apply it to a variety of applications.