Partial Fraction Decomposition Of (16x + 94) / (x^2 + 12x + 35)

In the realm of algebra and calculus, the decomposition of rational expressions into simpler fractions is a fundamental technique. This process, known as partial fraction decomposition, allows us to break down complex rational functions into a sum of simpler fractions, making them easier to integrate, differentiate, or analyze. This article will delve into the step-by-step process of finding the partial fraction decomposition for the rational expression (16x + 94) / (x^2 + 12x + 35). We will explore the underlying principles, demonstrate the practical application of the method, and highlight the importance of this technique in various mathematical contexts. Partial fraction decomposition is particularly useful when dealing with integrals of rational functions, as it transforms a complex integral into a sum of simpler integrals that can be readily evaluated. Furthermore, this technique is widely applied in solving differential equations, analyzing electrical circuits, and in various engineering and physics problems. Understanding partial fraction decomposition equips students and professionals alike with a powerful tool for simplifying and solving a wide range of mathematical problems. The key idea behind partial fraction decomposition is to express a rational function, which is a fraction where both the numerator and the denominator are polynomials, as a sum of fractions with simpler denominators. These simpler denominators are typically the factors of the original denominator. For instance, if the original denominator is a quadratic polynomial that can be factored into two distinct linear factors, then the partial fraction decomposition will involve two fractions, each with one of these linear factors as its denominator. The numerators of these simpler fractions are constants that need to be determined. The process of finding these constants involves algebraic manipulation, such as equating coefficients or substituting specific values for the variable. In more complex cases, the denominator might have repeated factors or irreducible quadratic factors, leading to a slightly more involved decomposition process. However, the fundamental principle remains the same: to break down a complex rational function into a sum of simpler fractions that are easier to work with. The ability to perform partial fraction decomposition is a crucial skill for anyone working with rational functions, and this article aims to provide a clear and comprehensive guide to this important technique.

Step 1: Factor the Denominator

The initial step in performing partial fraction decomposition is to factor the denominator of the rational expression. This process transforms the denominator into a product of simpler polynomials, which are essential for setting up the partial fractions. In our case, the denominator is the quadratic expression x^2 + 12x + 35. Factoring this quadratic involves finding two numbers that multiply to 35 and add up to 12. These numbers are 5 and 7. Therefore, we can rewrite the denominator as (x + 5)(x + 7). This factorization is crucial because it reveals the linear factors that will form the denominators of our partial fractions. The factored form of the denominator provides a clear roadmap for how the original rational expression can be decomposed. Each linear factor corresponds to a term in the partial fraction decomposition. For instance, the factor (x + 5) will give rise to a term of the form A / (x + 5), where A is a constant that we need to determine. Similarly, the factor (x + 7) will give rise to a term of the form B / (x + 7), where B is another constant to be found. The goal is to express the original rational expression as the sum of these simpler fractions. Factoring the denominator is not always straightforward, especially for higher-degree polynomials. However, for quadratic expressions, there are several techniques available, such as factoring by grouping, using the quadratic formula, or recognizing perfect square trinomials. The ability to factor polynomials efficiently is a valuable skill in algebra and is essential for many mathematical operations, including partial fraction decomposition. Once the denominator is factored, we can proceed to the next step, which involves setting up the partial fraction decomposition. This step involves writing the original rational expression as the sum of fractions with the factored denominators and unknown constants in the numerators. The values of these constants are then determined using algebraic techniques, such as equating coefficients or substituting specific values for the variable. The factored denominator serves as the foundation for the entire decomposition process, making it a critical first step.

Factoring x^2 + 12x + 35

To factor the quadratic expression x^2 + 12x + 35, we need to find two numbers that multiply to 35 (the constant term) and add up to 12 (the coefficient of the x term). These numbers are 5 and 7, since 5 * 7 = 35 and 5 + 7 = 12. Therefore, we can factor the quadratic as (x + 5)(x + 7). This factorization is the key to the partial fraction decomposition, as it provides the denominators for the simpler fractions.

So, we have:

rac{16x + 94}{x^2 + 12x + 35} = rac{16x + 94}{(x + 5)(x + 7)}

Step 2: Set Up the Partial Fraction Decomposition

Now that we have factored the denominator, the next step is to set up the partial fraction decomposition. This involves expressing the original rational expression as the sum of simpler fractions, each with one of the factors of the denominator as its denominator. In our case, the denominator is (x + 5)(x + 7), which means we will have two partial fractions: one with (x + 5) as the denominator and another with (x + 7) as the denominator. The numerators of these fractions will be constants, which we will denote as A and B. Thus, we can write the partial fraction decomposition as follows:

rac{16x + 94}{(x + 5)(x + 7)} = rac{A}{x + 5} + rac{B}{x + 7}

This equation represents the fundamental structure of the partial fraction decomposition. Our goal now is to find the values of the constants A and B. These constants are crucial because they determine the specific form of the partial fractions. The process of finding these constants involves algebraic manipulation, such as clearing the denominators and equating coefficients or substituting specific values for x. The setup of the partial fraction decomposition is based on the principle that any rational expression with a denominator that can be factored into distinct linear factors can be expressed as the sum of fractions with those factors as denominators. This principle is a cornerstone of partial fraction decomposition and is essential for solving a wide range of mathematical problems. The number of partial fractions in the decomposition is equal to the number of distinct linear factors in the denominator. If there are repeated linear factors, the setup becomes slightly more complex, involving terms with increasing powers of the repeated factor in the denominator. However, in our case, we have two distinct linear factors, making the setup relatively straightforward. The next step in the process is to solve for the constants A and B. This can be done using various algebraic techniques, and the choice of technique often depends on the specific form of the rational expression. Once the constants are found, we will have the complete partial fraction decomposition, which can then be used for further analysis or computation.

Step 3: Solve for the Constants

To find the values of the constants A and B, we need to eliminate the denominators in our equation:

rac{16x + 94}{(x + 5)(x + 7)} = rac{A}{x + 5} + rac{B}{x + 7}

We can do this by multiplying both sides of the equation by the common denominator, which is (x + 5)(x + 7). This gives us:

$16x + 94 = A(x + 7) + B(x + 5)$

Now, we have a couple of methods to solve for A and B:

Method 1: Substitution

We can substitute specific values of x that will eliminate one of the constants, allowing us to solve for the other. Let's start by setting x = -5. This will eliminate the term with B:

$16(-5) + 94 = A(-5 + 7) + B(-5 + 5)$

$-80 + 94 = 2A + 0$

$14 = 2A$

$A = 7$

Now, let's set x = -7. This will eliminate the term with A:

$16(-7) + 94 = A(-7 + 7) + B(-7 + 5)$

$-112 + 94 = 0 + B(-2)$

$-18 = -2B$

$B = 9$

Method 2: Equating Coefficients

We can expand the equation and group like terms:

$16x + 94 = Ax + 7A + Bx + 5B$

$16x + 94 = (A + B)x + (7A + 5B)$

Now, we can equate the coefficients of the corresponding terms. This gives us a system of two linear equations:

• Coefficient of x: $16 = A + B$
• Constant term: $94 = 7A + 5B$

We can solve this system of equations using substitution or elimination. From the first equation, we have $A = 16 - B$. Substituting this into the second equation:

$94 = 7(16 - B) + 5B$

$94 = 112 - 7B + 5B$

$94 = 112 - 2B$

$-18 = -2B$

$B = 9$

Now, substitute B = 9 back into $A = 16 - B$:

$A = 16 - 9$

$A = 7$

Both methods give us the same results: A = 7 and B = 9.

Step 4: Write the Partial Fraction Decomposition

Now that we have found the values of A and B, we can write the partial fraction decomposition:

rac{16x + 94}{x^2 + 12x + 35} = rac{7}{x + 5} + rac{9}{x + 7}

This is the partial fraction decomposition of the given rational expression. We have successfully broken down the original complex fraction into a sum of simpler fractions.

Conclusion: Applications and Significance

In conclusion, the partial fraction decomposition of the rational expression (16x + 94) / (x^2 + 12x + 35) is (7 / (x + 5)) + (9 / (x + 7)). This process demonstrates the power of algebraic manipulation in simplifying complex expressions. Partial fraction decomposition is not merely an algebraic exercise; it is a crucial technique with wide-ranging applications in calculus, differential equations, and engineering. For instance, when integrating rational functions, decomposing them into partial fractions often transforms a seemingly intractable integral into a sum of simpler integrals that can be easily evaluated using standard techniques. This is particularly important in fields like physics and engineering, where integrals of rational functions frequently arise in the context of solving differential equations and modeling physical systems. Furthermore, partial fraction decomposition plays a significant role in the analysis of electrical circuits, where it is used to determine the response of a circuit to different input signals. In control systems engineering, this technique is employed to analyze the stability and performance of feedback control systems. The ability to decompose rational expressions into simpler fractions provides a deeper understanding of the underlying mathematical structure and facilitates the solution of complex problems. The process we have followed, from factoring the denominator to solving for the constants and writing the final decomposition, highlights the systematic approach required for this technique. Each step is crucial, and a mistake in any step can lead to an incorrect result. Therefore, a thorough understanding of the underlying principles and careful execution of each step are essential. The significance of partial fraction decomposition extends beyond its immediate applications in mathematics. It exemplifies the broader mathematical strategy of breaking down complex problems into simpler, more manageable parts. This approach is a cornerstone of problem-solving in many areas of science and engineering, and mastering partial fraction decomposition provides valuable insights into this fundamental strategy. The importance of this technique cannot be overstated, as it provides a bridge between algebraic manipulation and advanced mathematical concepts. By mastering partial fraction decomposition, students and professionals alike gain a powerful tool for tackling a wide range of problems in mathematics and its applications.