Partial Fraction Decomposition Of (9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35)

JU07/22/2025 08, 2025 by THE IDEN 80 views

Decomposing Rational Expressions into Partial Fractions: A Comprehensive Guide

In the realm of algebra and calculus, partial fraction decomposition stands as a powerful technique to simplify complex rational expressions. A rational expression, essentially a fraction where the numerator and denominator are polynomials, can sometimes be unwieldy to work with directly. Partial fraction decomposition allows us to break down these complex fractions into simpler, more manageable components. This article delves into the intricacies of this method, providing a step-by-step guide along with illustrative examples.

The essence of partial fraction decomposition lies in expressing a given rational function as a sum of simpler fractions, each having a denominator that is a factor of the original denominator. This process is particularly useful when dealing with integration, as integrating simpler fractions is often significantly easier than integrating the original complex fraction. Moreover, it finds applications in various fields such as solving differential equations, analyzing electrical circuits, and even in certain areas of engineering and physics.

Understanding Rational Expressions

Before diving into the decomposition process, it's crucial to have a firm grasp of what rational expressions are. A rational expression is simply a fraction where both the numerator and denominator are polynomials. For example, (x^2 + 3x + 2) / (x - 1) and (5x^3 - 7) / (x^2 + 4) are both rational expressions. The degree of a polynomial is the highest power of the variable in the expression. The degree of the numerator and denominator plays a vital role in determining the approach to partial fraction decomposition.

Rational expressions can be classified into two main categories: proper and improper. A rational expression is considered proper if the degree of the numerator is less than the degree of the denominator. Conversely, if the degree of the numerator is greater than or equal to the degree of the denominator, the expression is deemed improper. This distinction is critical because the decomposition process differs slightly depending on whether the expression is proper or improper.

Steps for Partial Fraction Decomposition

The process of partial fraction decomposition involves a series of well-defined steps. These steps ensure a systematic approach to breaking down complex rational expressions. Let's outline these steps in detail:

1. Check if the Rational Expression is Proper: The first step is to determine whether the given rational expression is proper or improper. As mentioned earlier, this depends on the degrees of the numerator and the denominator. If the expression is improper, proceed to the next step; otherwise, skip to step 3.

2. Perform Long Division (if Improper): If the rational expression is improper, we need to perform polynomial long division. This process divides the numerator by the denominator, resulting in a quotient and a remainder. The original expression can then be rewritten as the sum of the quotient and a new rational expression where the numerator is the remainder and the denominator is the original denominator. This new rational expression will always be proper.

3. Factor the Denominator: The next crucial step is to factor the denominator of the rational expression completely. Factoring involves expressing the denominator as a product of simpler polynomials, typically linear (degree 1) or quadratic (degree 2) factors. The nature of these factors dictates the form of the partial fractions.

4. Set up the Partial Fraction Decomposition: Based on the factored denominator, we set up the partial fraction decomposition. This involves expressing the original rational expression as a sum of fractions, each with a denominator corresponding to one of the factors of the original denominator. The numerators of these partial fractions are unknowns that we need to determine. The form of the numerators depends on the type of factors in the denominator:

Linear Factors: For each linear factor of the form (ax + b), we include a partial fraction of the form A / (ax + b), where A is a constant to be determined.
Repeated Linear Factors: If a linear factor (ax + b) appears multiple times (say, n times), we include n partial fractions of the form A1 / (ax + b) + A2 / (ax + b)^2 + ... + An / (ax + b)^n, where A1, A2, ..., An are constants to be determined.
Irreducible Quadratic Factors: For each irreducible quadratic factor (ax^2 + bx + c) (a quadratic that cannot be factored into linear factors with real coefficients), we include a partial fraction of the form (Ax + B) / (ax^2 + bx + c), where A and B are constants to be determined.
Repeated Irreducible Quadratic Factors: If an irreducible quadratic factor (ax^2 + bx + c) appears multiple times (say, n times), we include n partial fractions of the form (A1x + B1) / (ax^2 + bx + c) + (A2x + B2) / (ax^2 + bx + c)^2 + ... + (Anx + Bn) / (ax^2 + bx + c)^n, where A1, B1, A2, B2, ..., An, Bn are constants to be determined.

5. Solve for the Unknown Constants: Once we have set up the partial fraction decomposition, the next step is to solve for the unknown constants in the numerators. There are two common methods for doing this:

Method of Clearing Denominators: This method involves multiplying both sides of the equation by the original denominator. This eliminates the fractions, resulting in a polynomial equation. We can then solve for the constants by equating the coefficients of like terms on both sides of the equation.
Method of Substituting Values: This method involves substituting specific values for the variable (typically the roots of the factors in the denominator) into the equation. Each substitution yields an equation involving the unknown constants. By substituting a sufficient number of values, we can create a system of equations that can be solved for the constants.

6. Write the Partial Fraction Decomposition: After determining the values of the unknown constants, we substitute them back into the partial fraction decomposition setup in step 4. This gives us the final decomposition of the original rational expression.

Example: Decomposing a Rational Expression

Let's illustrate the process with an example. Consider the rational expression:

(9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35)

1. Check if Proper: The degree of the numerator (3) is greater than the degree of the denominator (2), so the expression is improper.

2. Perform Long Division: Dividing (9x^3 + 10x^2 - 326x + 207) by (x^2 + 2x - 35) gives a quotient of (9x - 8) and a remainder of (-9x + 27). Thus, we can rewrite the expression as:

(9x - 8) + (-9x + 27) / (x^2 + 2x - 35)

3. Factor the Denominator: The denominator factors as (x + 7)(x - 5).

4. Set up the Partial Fraction Decomposition: Since we have two distinct linear factors, the partial fraction decomposition is of the form:

(-9x + 27) / ((x + 7)(x - 5)) = A / (x + 7) + B / (x - 5)

5. Solve for the Constants: Let's use the method of clearing denominators. Multiplying both sides by (x + 7)(x - 5) gives:

-9x + 27 = A(x - 5) + B(x + 7)

Expanding the right side, we get:

-9x + 27 = Ax - 5A + Bx + 7B

Grouping like terms:

-9x + 27 = (A + B)x + (-5A + 7B)

Equating coefficients, we get the following system of equations:

A + B = -9
-5A + 7B = 27

Solving this system, we find A = -8 and B = -1.

6. Write the Partial Fraction Decomposition: Substituting the values of A and B, we get:

(-9x + 27) / ((x + 7)(x - 5)) = -8 / (x + 7) + (-1) / (x - 5)

Therefore, the partial fraction decomposition of the original expression is:

(9x - 8) - 8 / (x + 7) - 1 / (x - 5)

Applications of Partial Fraction Decomposition

Partial fraction decomposition is not merely a mathematical exercise; it has significant practical applications in various fields. One of its most prominent uses is in calculus, particularly in the integration of rational functions. As mentioned earlier, integrating simpler fractions obtained through decomposition is often much easier than integrating the original complex fraction. This is especially true when dealing with rational functions that arise in real-world problems.

In engineering, partial fraction decomposition finds applications in areas such as circuit analysis and control systems. Electrical circuits, for instance, can be modeled using differential equations, and the solutions to these equations often involve rational functions. Decomposing these functions into partial fractions simplifies the analysis and allows engineers to understand the behavior of the circuit more effectively.

Furthermore, partial fraction decomposition is a valuable tool in solving differential equations. Many differential equations that arise in physics and engineering involve rational functions. By decomposing these functions, we can often find closed-form solutions to the differential equations, providing valuable insights into the physical systems they describe.

Common Mistakes to Avoid

While partial fraction decomposition is a powerful technique, it's essential to be aware of common mistakes that can occur during the process. Avoiding these pitfalls ensures accurate and efficient decomposition.

One common mistake is failing to check if the rational expression is proper. Attempting to decompose an improper fraction directly without performing long division will lead to incorrect results. Always ensure that the degree of the numerator is less than the degree of the denominator before proceeding with the decomposition.

Another frequent error is incorrectly factoring the denominator. Accurate factoring is crucial for setting up the partial fraction decomposition correctly. Double-check your factoring to ensure that you have identified all the factors and their multiplicities.

A third mistake is setting up the partial fractions incorrectly. The form of the partial fractions depends on the type of factors in the denominator. Make sure you include the appropriate terms for each linear, repeated linear, irreducible quadratic, and repeated irreducible quadratic factor.

Finally, errors can occur when solving for the unknown constants. Whether you use the method of clearing denominators or the method of substituting values, pay close attention to the algebraic manipulations and ensure that you solve the system of equations accurately.

Conclusion

Partial fraction decomposition is a fundamental technique in algebra and calculus that allows us to simplify complex rational expressions. By breaking down these expressions into simpler fractions, we can make them more manageable for integration, solving differential equations, and other applications. This comprehensive guide has provided a step-by-step approach to partial fraction decomposition, along with illustrative examples and common mistakes to avoid. Mastering this technique will undoubtedly enhance your problem-solving skills in mathematics and related fields.

Decomposing (9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35) into Partial Fractions

Let's apply the principles of partial fraction decomposition to the specific rational expression provided: (9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35). This example will solidify your understanding of the process and highlight the key steps involved.

Step 1: Check if the Rational Expression is Proper

The first crucial step is to determine whether the given rational expression is proper or improper. A rational expression is considered proper if the degree of the numerator is less than the degree of the denominator. In our case, the numerator is 9x^3 + 10x^2 - 326x + 207, which has a degree of 3 (the highest power of x). The denominator is x^2 + 2x - 35, which has a degree of 2. Since the degree of the numerator (3) is greater than the degree of the denominator (2), the rational expression is improper.

This is a critical observation because the procedure for decomposing improper rational expressions differs slightly from that for proper expressions. Specifically, we need to perform polynomial long division before we can proceed with partial fraction decomposition.

Step 2: Perform Polynomial Long Division

Since the rational expression is improper, we must perform polynomial long division to divide the numerator (9x^3 + 10x^2 - 326x + 207) by the denominator (x^2 + 2x - 35). This process will yield a quotient and a remainder. The quotient represents the polynomial part of the expression, while the remainder will form the numerator of a new, proper rational expression.

Performing the long division, we obtain:

 9x - 8
x^2 + 2x - 35 | 9x^3 + 10x^2 - 326x + 207
 -(9x^3 + 18x^2 - 315x)
 ------------------------
 -8x^2 - 11x + 207
 -(-8x^2 - 16x + 280)
 ------------------------
 5x - 73

From the long division, we find that the quotient is 9x - 8 and the remainder is 5x - 73. Therefore, we can rewrite the original improper rational expression as the sum of the quotient and a proper rational expression:

(9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35) = (9x - 8) + (5x - 73) / (x^2 + 2x - 35)

Now, we can focus on decomposing the proper rational expression (5x - 73) / (x^2 + 2x - 35) into partial fractions.

Step 3: Factor the Denominator

The next step is to factor the denominator of the proper rational expression, which is x^2 + 2x - 35. Factoring this quadratic expression involves finding two numbers that multiply to -35 and add up to 2. These numbers are 7 and -5. Thus, the denominator can be factored as:

x^2 + 2x - 35 = (x + 7)(x - 5)

This factorization is crucial because it determines the form of the partial fraction decomposition. Since we have two distinct linear factors, (x + 7) and (x - 5), we will have two partial fractions, each with one of these factors as the denominator.

Step 4: Set up the Partial Fraction Decomposition

Based on the factored denominator, we can now set up the partial fraction decomposition. Since we have two distinct linear factors, (x + 7) and (x - 5), the decomposition will take the following form:

(5x - 73) / ((x + 7)(x - 5)) = A / (x + 7) + B / (x - 5)

where A and B are constants that we need to determine. These constants represent the numerators of the partial fractions. The goal is to find the values of A and B that make this equation true for all values of x (except for x = -7 and x = 5, where the denominators would be zero).

This setup is the heart of the partial fraction decomposition process. It expresses the original complex fraction as a sum of simpler fractions, each with a denominator that is a factor of the original denominator. The constants A and B are the unknowns that we need to solve for.

Step 5: Solve for the Unknown Constants

To find the values of the constants A and B, we can use either the method of clearing denominators or the method of substituting values. Let's use the method of clearing denominators in this example. This method involves multiplying both sides of the equation by the common denominator, which is (x + 7)(x - 5). This will eliminate the fractions and give us a polynomial equation.

Multiplying both sides of the equation (5x - 73) / ((x + 7)(x - 5)) = A / (x + 7) + B / (x - 5) by (x + 7)(x - 5), we get:

5x - 73 = A(x - 5) + B(x + 7)

Now, we expand the right side of the equation:

5x - 73 = Ax - 5A + Bx + 7B

Next, we group the terms with x and the constant terms:

5x - 73 = (A + B)x + (-5A + 7B)

For this equation to hold true for all values of x, the coefficients of the corresponding terms on both sides must be equal. This gives us a system of two linear equations in two unknowns:

A + B = 5 (equating the coefficients of x)
-5A + 7B = -73 (equating the constant terms)

We can solve this system of equations using various methods, such as substitution or elimination. Let's use the method of elimination. Multiply the first equation by 5 to get:

5A + 5B = 25

Now, add this equation to the second equation:

(5A + 5B) + (-5A + 7B) = 25 + (-73)

This simplifies to:

12B = -48

Dividing both sides by 12, we get:

B = -4

Now that we have the value of B, we can substitute it back into the first equation (A + B = 5) to find A:

A + (-4) = 5

Adding 4 to both sides, we get:

A = 9

Therefore, we have found the values of the constants: A = 9 and B = -4.

Step 6: Write the Partial Fraction Decomposition

Now that we have determined the values of the constants A and B, we can substitute them back into the partial fraction decomposition setup in step 4. This gives us the final decomposition of the proper rational expression:

(5x - 73) / ((x + 7)(x - 5)) = 9 / (x + 7) + (-4) / (x - 5)

Finally, we need to combine this decomposition with the quotient we obtained from the long division in step 2. Recall that we had:

(9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35) = (9x - 8) + (5x - 73) / (x^2 + 2x - 35)

Substituting the partial fraction decomposition of (5x - 73) / (x^2 + 2x - 35), we get:

(9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35) = (9x - 8) + 9 / (x + 7) - 4 / (x - 5)

This is the complete partial fraction decomposition of the original improper rational expression. We have successfully broken it down into a sum of simpler terms: a polynomial (9x - 8) and two proper rational fractions with linear denominators.

Summary of the Decomposition

In summary, the partial fraction decomposition of the rational expression (9x^3 + 10x^2 - 326x + 207) / (x^2 + 2x - 35) is:

(9x - 8) + 9 / (x + 7) - 4 / (x - 5)

This decomposition expresses the original complex rational expression as a sum of simpler terms, which can be useful for various applications, such as integration, solving differential equations, and analyzing systems in engineering and physics.

This example provides a detailed walkthrough of the partial fraction decomposition process, highlighting each step and the reasoning behind it. By understanding these steps, you can confidently decompose a wide range of rational expressions into partial fractions.