Partial Fraction Decomposition Of (7x + 18) / (x^2 + 9x)

Partial fraction decomposition is a crucial technique in algebra and calculus, particularly when dealing with integration and solving differential equations. Partial fraction decomposition allows us to break down a complex rational expression into simpler fractions that are easier to work with. In essence, it's the reverse process of adding fractions with different denominators. When faced with a rational expression where the degree of the numerator is less than the degree of the denominator, and the denominator can be factored, partial fraction decomposition becomes a powerful tool.

To truly grasp the essence of partial fraction decomposition, one must understand its fundamental principles. The process begins by factoring the denominator of the given rational expression. The nature of these factors—whether they are linear, repeated linear, irreducible quadratic, or repeated irreducible quadratic—dictates the form of the decomposition. Each distinct factor in the denominator corresponds to one or more simpler fractions in the decomposition. The numerators of these simpler fractions are constants or linear expressions, depending on the degree of the factor in the original denominator. By expressing a complex rational function as a sum of simpler fractions, we can often simplify complex calculations, such as finding integrals or inverse Laplace transforms. This technique is not just a mathematical trick but a fundamental tool in simplifying and solving a wide range of problems across mathematics, engineering, and physics.

Before diving into the specific problem, let's clarify the general procedure. First, ensure that the degree of the polynomial in the numerator is strictly less than the degree of the polynomial in the denominator. If not, perform polynomial long division to obtain a proper fraction. Next, factor the denominator completely. The factors will determine the structure of the partial fraction decomposition. For each linear factor (x - a), include a term of the form A / (x - a). For each repeated linear factor (x - a)^n, include terms of the form A_1 / (x - a) + A_2 / (x - a)^2 + ... + A_n / (x - a)^n. For each irreducible quadratic factor (ax^2 + bx + c), include a term of the form (Ax + B) / (ax^2 + bx + c). For each repeated irreducible quadratic factor (ax^2 + bx + c)^n, include terms of the form (A_1x + B_1) / (ax^2 + bx + c) + (A_2x + B_2) / (ax^2 + bx + c)^2 + ... + (A_nx + B_n) / (ax^2 + bx + c)^n. The constants A, B, A_i, and B_i are then determined by equating the numerators and solving the resulting system of equations. Understanding these general rules is crucial for successful application of partial fraction decomposition in various mathematical contexts.

Our task is to determine the correct partial fraction decomposition for the expression (7x + 18) / (x^2 + 9x). The first crucial step in this process is to factor the denominator. The denominator, x^2 + 9x, can be easily factored by taking out the common factor of x. This gives us x(x + 9). Now that we have factored the denominator, we can see that it consists of two distinct linear factors: x and (x + 9). According to the principles of partial fraction decomposition, each distinct linear factor in the denominator corresponds to a term in the decomposition with a constant numerator.

Given that we have two distinct linear factors, x and (x + 9), we can set up the partial fraction decomposition as follows: A / x + B / (x + 9), where A and B are constants that we need to determine. This form reflects the fact that each linear factor contributes a fraction with a constant numerator. The constant A corresponds to the factor x, and the constant B corresponds to the factor (x + 9). Our goal now is to find the values of A and B that make this decomposition equal to the original expression. This involves algebraic manipulation and solving a system of equations. The correct setup of the partial fraction decomposition is the foundation for accurately solving the problem. Errors in this initial step can lead to incorrect results, so it's essential to understand the underlying principles and apply them carefully.

Now, let's examine the given options in light of this understanding. Option A, A / x^2 + B / (9x), is incorrect because it incorrectly represents the factors in the denominator. The x^2 term suggests a repeated factor, which is not present in the original denominator once it is correctly factored. Option C, (Ax + B) / x^2 + C / (9x), is also incorrect for the same reason. It introduces a more complex numerator (Ax + B) for the x^2 term, which is not necessary for a simple linear factor like x. Option D, (Ax + B) / x + C / (x + 9), is incorrect because it uses a linear expression (Ax + B) in the numerator over the single factor x, which is an overcomplication for a linear factor. The correct form should only have constant numerators for linear factors. Therefore, the correct form of the partial fraction decomposition should have constant numerators over each of the distinct linear factors, which leads us to the correct answer, option B.

Based on our analysis, the correct form of the partial fraction decomposition for the expression (7x + 18) / (x^2 + 9x) is B. A / x + B / (x + 9). This form accurately represents the decomposition based on the distinct linear factors in the denominator. The other options present incorrect forms that do not align with the principles of partial fraction decomposition. To further solidify this understanding, let's briefly discuss why the other options are incorrect.

Option A, A / x^2 + B / (9x), is flawed because it misinterprets the factored form of the denominator. While it includes terms related to x, the presence of x^2 suggests a repeated factor, which is not the case. When we factored the denominator x^2 + 9x, we obtained x(x + 9), which has two distinct linear factors, not a repeated factor of x. Therefore, using x^2 in the denominator of one of the partial fractions is incorrect. Option C, (Ax + B) / x^2 + C / (9x), suffers from a similar issue. It also includes the x^2 term, and additionally, it uses a linear expression (Ax + B) in the numerator over x^2. This form is more complex than necessary and does not correctly represent the partial fraction decomposition for the given expression. Option D, (Ax + B) / x + C / (x + 9), is incorrect because it places a linear expression (Ax + B) over the linear factor x. According to the rules of partial fraction decomposition, when the denominator has distinct linear factors, the numerators should be constants, not linear expressions.

In summary, the key to correctly identifying the form of the partial fraction decomposition lies in accurately factoring the denominator and applying the rules for decomposition based on the types of factors present. For distinct linear factors, we use constant numerators. For repeated linear factors, we include terms with increasing powers of the factor in the denominator. For irreducible quadratic factors, we use linear expressions in the numerator, and so on. By carefully applying these rules, we can correctly set up the partial fraction decomposition and proceed with solving for the unknown constants. The form A / x + B / (x + 9) perfectly fits this scenario, making it the correct choice for the given expression. This correct setup is crucial for the subsequent steps of solving for A and B and ultimately integrating or otherwise manipulating the expression.

In conclusion, the correct partial fraction decomposition form for the expression (7x + 18) / (x^2 + 9x) is B. A / x + B / (x + 9). This result is obtained by correctly factoring the denominator into distinct linear factors and applying the fundamental principles of partial fraction decomposition. Understanding these principles is essential for simplifying complex rational expressions and solving a variety of problems in mathematics and engineering.

Partial fraction decomposition is not just a mathematical technique; it is a problem-solving tool that bridges algebraic manipulation with calculus and differential equations. By mastering this technique, students and professionals alike gain a powerful method for tackling complex problems. The ability to decompose rational functions into simpler parts allows for easier integration, simplifies the process of finding inverse Laplace transforms, and aids in solving systems of differential equations. The correct application of partial fraction decomposition requires a solid understanding of algebra, particularly factoring, and a methodical approach to setting up and solving the resulting equations.

The process of finding the constants A and B, once the correct form is established, involves clearing the denominators and equating the coefficients of like terms. This leads to a system of linear equations that can be solved using various methods, such as substitution, elimination, or matrix techniques. The values of these constants then complete the partial fraction decomposition, allowing for further analysis or manipulation of the original expression. This entire process underscores the interconnectedness of different areas of mathematics and highlights the importance of a strong foundation in fundamental concepts. The mastery of partial fraction decomposition opens doors to more advanced topics and real-world applications, making it a valuable skill for anyone pursuing careers in science, technology, engineering, and mathematics.