Partial Fraction Decomposition is a method used to express a rational function (a fraction of two polynomials) as a sum of simpler fractions. This technique is especially useful in calculus and algebra for simplifying integration or solving differential equations. In this article, we will explain how partial fraction decomposition works, provide examples, and introduce a Partial Fraction Decomposition Calculator to automate the process.

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What is Partial Fraction Decomposition?

Partial fraction decomposition is a way of breaking down a complex rational function into simpler components, called partial fractions, that can be more easily handled in algebraic operations like integration, differentiation, or solving equations.

For example, if you have a rational function like:

(5x + 7) / (x² – 4)

You can decompose it into simpler fractions, such as:

(A / (x – 2)) + (B / (x + 2))

Where A and B are constants to be determined.

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Steps for Partial Fraction Decomposition

The process of partial fraction decomposition depends on the form of the denominator. Here’s how it works step-by-step:

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Step 1: Factor the Denominator

Before performing partial fraction decomposition, the denominator must be factored as much as possible.

For example:

• (x² – 4) factors into (x – 2)(x + 2).
• (x³ – 3x² + 2x) factors into x(x – 1)(x – 2).

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Step 2: Set Up the Partial Fractions

Once you’ve factored the denominator, you can set up the partial fractions. The form of the partial fractions depends on the type of factors in the denominator.

1. If the denominator has linear factors (like x – a), use a fraction of the form A / (x – a).
2. If the denominator has repeated linear factors (like (x – a)²), use a fraction for each power of the factor: A / (x – a) and B / (x – a)².
3. If the denominator has irreducible quadratic factors (like x² + bx + c), use a fraction of the form (Ax + B) / (x² + bx + c).

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Step 3: Solve for the Constants

After setting up the partial fractions, you’ll solve for the constants (such as A, B, etc.) by multiplying both sides of the equation by the denominator of the left side (the original denominator) and then solving the resulting equation for the constants.

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Example 1: Simple Partial Fraction Decomposition

Let’s break down a simple rational function:

(5x + 7) / (x² – 4)

1. Factor the Denominator:
   • The denominator x² – 4 factors into (x – 2)(x + 2).
2. Set up the Partial Fractions:
   • We set up two fractions: A / (x – 2) and B / (x + 2).
   So, we have: (5x + 7) / (x² – 4) = A / (x – 2) + B / (x + 2)
3. Multiply both sides by (x – 2)(x + 2): 5x + 7 = A(x + 2) + B(x – 2)
4. Expand and group like terms: 5x + 7 = A(x) + 2A + B(x) – 2BCombine like terms: 5x + 7 = (A + B)x + (2A – 2B)
5. Set up the system of equations: Compare the coefficients of x and the constant terms:
   • A + B = 5
   • 2A – 2B = 7
6. Solve the system: Solving this system, we get:
   • A = 6
   • B = -1

So, the partial fraction decomposition is:

(5x + 7) / (x² – 4) = 6 / (x – 2) – 1 / (x + 2)

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Example 2: Decomposition with Repeated Linear Factors

Let’s try a more complex example:

(2x + 3) / (x²(x + 1))

1. Factor the Denominator:
   • The denominator is already factored: x²(x + 1).
2. Set up the Partial Fractions:
   • We set up three fractions: A / x, B / x², and C / (x + 1).
   So, we have: (2x + 3) / (x²(x + 1)) = A / x + B / x² + C / (x + 1)
3. Multiply both sides by x²(x + 1): 2x + 3 = A(x)(x + 1) + B(x + 1) + C(x²)
4. Expand and group like terms: 2x + 3 = A(x² + x) + B(x + 1) + Cx²Combine like terms: 2x + 3 = (A + C)x² + (A + B)x + B
5. Set up the system of equations: Compare the coefficients:
   • A + C = 0 (coefficient of x²)
   • A + B = 2 (coefficient of x)
   • B = 3 (constant term)
6. Solve the system: Solving this system, we get:
   • A = -3
   • B = 3
   • C = 3

So, the partial fraction decomposition is:

(2x + 3) / (x²(x + 1)) = -3 / x + 3 / x² + 3 / (x + 1)

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Partial Fraction Decomposition Calculator

Instead of manually solving each decomposition, you can use a Partial Fraction Decomposition Calculator to automate the process. These online tools let you input a rational function and quickly calculate its partial fraction decomposition. Here’s how to use it:

1. Enter the Numerator and Denominator:
   • Input the numerator and denominator of the rational function you want to decompose.
2. Select the Type of Function:
   • Indicate whether the denominator contains simple factors, repeated factors, or irreducible quadratics.
3. Calculate:
   • Click the “Calculate” button, and the tool will provide the partial fraction decomposition.

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Frequently Asked Questions (FAQ)

1. What is Partial Fraction Decomposition used for?

Partial fraction decomposition is commonly used in calculus, especially for simplifying integration. It is also useful in solving differential equations, evaluating integrals, and simplifying complex algebraic expressions.

2. Can all rational functions be decomposed?

Yes, any rational function (a ratio of two polynomials) can be decomposed into partial fractions as long as the denominator is factored completely.

3. When do I use repeated factors in partial fraction decomposition?

When the denominator contains repeated factors, such as (x – 1)², you need to use a fraction for each power of the factor. For example, you would write A / (x – 1) and B / (x – 1)².

4. Can I decompose a fraction with irreducible quadratics?

Yes, when the denominator contains irreducible quadratic factors (like x² + 1), you use fractions of the form (Ax + B) / (x² + 1) for each quadratic factor.

5. How do I know which method to use?

• If the denominator factors into linear terms (like (x – a)), you use simple fractions like A / (x – a).
• If the denominator has repeated factors (like (x – a)²), you include fractions for each power of the factor.
• If the denominator has irreducible quadratic factors (like x² + 1), you use (Ax + B) / (x² + 1).

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Conclusion

Partial fraction decomposition is a powerful technique for simplifying rational functions into simpler fractions. This method is widely used in calculus, algebra, and differential equations, especially for simplifying integrals and solving complex equations. Whether you’re working with simple linear factors or more complex repeated and irreducible factors, the Partial Fraction Decomposition Calculator can help you quickly and accurately decompose any rational function.