Partial Fraction Decomposition Of (9x^2-8)/(3x^2+4) A Comprehensive Guide

Partial fraction decomposition is a powerful technique in mathematics, particularly in calculus and algebra, used to break down rational functions into simpler fractions. This decomposition simplifies the process of integrating complex rational functions and is crucial in various mathematical applications. At its core, partial fraction decomposition is the reverse process of adding fractions with different denominators. It's like taking a combined fraction and separating it back into its original components. This method is particularly useful when dealing with rational expressions where the degree of the numerator is less than the degree of the denominator. However, even if the degree of the numerator is greater than or equal to the denominator, we can perform polynomial long division first to obtain a proper fraction, which can then be decomposed. The essence of this technique lies in expressing a complicated rational function as a sum of simpler fractions, each with a denominator that is a factor of the original denominator. This simplifies complex algebraic manipulations and makes integration and other operations more manageable. The process involves several key steps, including factoring the denominator, setting up the partial fraction decomposition, and solving for the unknown coefficients. The complexity of the decomposition depends on the nature of the factors in the denominator, whether they are linear, quadratic, repeated, or irreducible. Understanding the nuances of each case is essential for mastering this technique.

Let's delve into the heart of the matter: the expression (9x^2-8)/(3x2+4). Our goal is to determine the correct form of its partial fraction decomposition. Before diving into the options, it's crucial to understand the nature of the denominator, 3x^2+4. This is a quadratic expression, and more specifically, an irreducible quadratic expression. An irreducible quadratic expression cannot be factored further into linear factors with real coefficients. This characteristic significantly influences the form of the partial fraction decomposition. When dealing with irreducible quadratic factors in the denominator, the numerator of the corresponding partial fraction will be a linear expression of the form Ax + B. This is a fundamental rule in partial fraction decomposition. Now, let's consider the given expression. The degree of the numerator (9x^2-8) is 2, and the degree of the denominator (3x^2+4) is also 2. Since the degrees are equal, we first need to perform polynomial long division to rewrite the expression as a quotient plus a proper fraction (where the degree of the numerator is less than the degree of the denominator). Dividing 9x^2-8 by 3x^2+4, we get a quotient of 3 and a remainder of -20. Thus, the expression can be rewritten as 3 - 20/(3x^2+4). Now, we focus on the fractional part, -20/(3x^2+4). Since the denominator is an irreducible quadratic, the partial fraction decomposition will have the form A/(3x^2+4), where A is a constant. This is because the numerator should be a polynomial of degree one less than the denominator.

Now, let's dissect the provided options to pinpoint the correct form of the partial fraction decomposition for our expression. Option A, (Ax+B)/(3x^2+4) + (Cx+D)/(3x²⁺⁴⁾2, suggests a scenario where the denominator has a repeated irreducible quadratic factor. However, our original expression's denominator, 3x^2+4, is not raised to any power, indicating it's not a repeated factor. Therefore, option A is not the correct form for our decomposition. It introduces unnecessary complexity that doesn't align with the structure of the original expression. Option B, A/(3x^2+4) + B/(3x²⁺⁴⁾2, also implies a repeated irreducible quadratic factor in the denominator. Similar to option A, this doesn't match the simple, non-repeated quadratic factor in our original expression. The presence of the (3x²⁺⁴⁾2 term suggests a more complex scenario than what we have. Thus, option B is also incorrect. This option adds an extra term that is not required for the decomposition of our expression. Option C is not a partial fraction decomposition but a discussion category, so it's not a valid answer. So, this question appears to have a mistake because the correct answer is not listed in the options. The correct form of the partial fraction decomposition for the expression (9x^2-8)/(3x2+4) is simply A/(3x^2+4), where A is a constant. This is because the denominator is an irreducible quadratic, and after performing polynomial long division, we are left with a proper fraction with this denominator.

Having analyzed the options, it's clear that neither A nor B accurately represents the partial fraction decomposition of (9x^2-8)/(3x2+4). The crucial step is recognizing that the given expression initially requires polynomial long division due to the equal degrees of the numerator and denominator. As we established earlier, dividing 9x^2-8 by 3x^2+4 yields 3 - 20/(3x^2+4). The focus then shifts to the fractional part, -20/(3x^2+4). Since 3x^2+4 is an irreducible quadratic, the correct partial fraction decomposition form should only have a single term with this denominator. The numerator of this term will be a constant because the degree of the numerator should be one less than the degree of the denominator. Therefore, the correct form is A/(3x^2+4). However, neither of the provided options matches this form. Options A and B both include an additional term with (3x²⁺⁴⁾2 in the denominator, which is not necessary for this particular expression. This highlights the importance of carefully analyzing the denominator and recognizing whether it's a repeated factor or not. In our case, 3x^2+4 is not a repeated factor, so we only need one term in the decomposition. The constant A can be determined by equating the original fraction with its decomposed form and solving for A. In this case, A would be -20.

To truly master partial fraction decomposition, it's essential to grasp the fundamental principles and apply them systematically. Here are some key takeaways to solidify your understanding. First and foremost, always check if the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first. This step is crucial for obtaining a proper fraction that can be decomposed. Next, factor the denominator completely. The factors will determine the form of the partial fraction decomposition. Pay close attention to whether the factors are linear, quadratic, repeated, or irreducible. For each linear factor (ax + b), include a term of the form A/(ax + b) in the decomposition. For each irreducible quadratic factor (ax^2 + bx + c), include a term of the form (Ax + B)/(ax^2 + bx + c). If a factor is repeated, include terms for each power of the factor. For example, if (ax + b)^2 is a factor, include both A/(ax + b) and B/(ax + b)^2. Once you've set up the decomposition, solve for the unknown coefficients (A, B, C, etc.). This usually involves multiplying both sides of the equation by the original denominator and then either substituting values for x or equating coefficients of like terms. Finally, always double-check your work by adding the partial fractions back together to see if you get the original fraction. This is a simple but effective way to catch any errors. By following these steps and practicing regularly, you'll become proficient in partial fraction decomposition and be able to tackle even the most complex rational functions with confidence. This technique is not just a mathematical exercise; it's a powerful tool that has wide-ranging applications in various fields, making it a valuable skill to acquire.