Partial Fraction Decomposition Of Rational Function 8x^4+3x^3-115x^2-39x+200

In the realm of mathematics, particularly in calculus and algebra, partial fraction decomposition stands as a powerful technique for simplifying complex rational functions. This method allows us to break down a complicated fraction into a sum of simpler fractions, making it easier to integrate, differentiate, or manipulate algebraically. In this comprehensive exploration, we will delve into the intricacies of partial fraction decomposition, focusing on a specific example to illustrate the process and the underlying principles. Our focus will be on the rational function ${\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} }$, and we will meticulously dissect it to determine the correct partial fraction decomposition. This article aims to provide a clear, step-by-step explanation, ensuring that readers gain a solid understanding of this essential mathematical tool.

At its core, partial fraction decomposition is the process of expressing a rational function as a sum of simpler fractions. A rational function is a function that can be written as the ratio of two polynomials, ${ P(x) }$ and ${ Q(x) }$, where ${ Q(x) }$ is not zero. The goal of partial fraction decomposition is to rewrite the rational function ${ \frac{P(x)}{Q(x)} }$ in the form:

$$\frac{P(x)}{Q(x)} = \frac{A_1}{Q_1(x)} + \frac{A_2}{Q_2(x)} + \cdots + \frac{A_n}{Q_n(x)}$$

where ${ Q_1(x), Q_2(x), \ldots, Q_n(x) }$ are factors of ${ Q(x) }$, and ${ A_1, A_2, \ldots, A_n }$ are constants or polynomials of lower degree than the corresponding denominators. The choice of the form of the partial fractions depends on the nature of the factors of the denominator ${ Q(x) }$.

To effectively apply partial fraction decomposition, it's crucial to recognize the different types of factors that might appear in the denominator and how to handle each one. Factors can be linear (e.g., ${ ax + b }$), repeated linear (e.g., ${ (ax + b)^n }$), irreducible quadratic (e.g., ${ ax^2 + bx + c }$ where the discriminant ${ b^2 - 4ac < 0 }$), or repeated irreducible quadratic (e.g., ${ (ax^2 + bx + c)^n }$). Each type of factor requires a specific form in the partial fraction decomposition. This foundational understanding is paramount for tackling more complex rational functions and ensuring accurate decomposition.

Now, let's turn our attention to the specific rational function we aim to decompose:

$$\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2}$$

This function has a denominator, ${ 3x(x^2 + 10)^2 }$, which consists of a linear factor, ${ 3x }$, and a repeated irreducible quadratic factor, ${ (x^2 + 10)^2 }$. Understanding the nature of these factors is crucial for setting up the correct form of the partial fraction decomposition. The linear factor ${ 3x }$ will contribute a term of the form ${ \frac{A}{3x} }$, where ${ A }$ is a constant. The repeated irreducible quadratic factor ${ (x^2 + 10)^2 }$ will contribute two terms: ${ \frac{Bx + C}{x^2 + 10} }$ and ${ \frac{Dx + E}{(x^2 + 10)^2} }$, where ${ B, C, D, }$ and ${ E }$ are constants. Therefore, the partial fraction decomposition of the given rational function will have the following form:

$$\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} = \frac{A}{3x} + \frac{Bx + C}{x^2 + 10} + \frac{Dx + E}{(x^2 + 10)^2}$$

This setup is the critical first step in the decomposition process. It correctly accounts for the structure of the denominator and ensures that we include all necessary terms in the partial fraction expansion. The next steps involve determining the values of the constants ${ A, B, C, D, }$ and ${ E }$, which we will address in the subsequent sections.

With the form of the partial fraction decomposition established, the next step is to determine the values of the constants ${ A, B, C, D, }$ and ${ E }$ in the equation:

$$\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} = \frac{A}{3x} + \frac{Bx + C}{x^2 + 10} + \frac{Dx + E}{(x^2 + 10)^2}$$

To find these constants, we first clear the denominators by multiplying both sides of the equation by the original denominator, ${ 3x(x^2 + 10)^2 }$. This gives us:

$$8 x^4 + 3 x^3 - 115 x^2 - 39 x + 200 = A(x^2 + 10)^2 + (Bx + C)(3x)(x^2 + 10) + (Dx + E)(3x)$$

Expanding the terms on the right side, we get:

$$8 x^4 + 3 x^3 - 115 x^2 - 39 x + 200 = A(x^4 + 20x^2 + 100) + (Bx + C)(3x^3 + 30x) + 3Dx^2 + 3Ex$$

$$8 x^4 + 3 x^3 - 115 x^2 - 39 x + 200 = Ax^4 + 20Ax^2 + 100A + 3Bx^4 + 30Bx^2 + 3Cx^3 + 30Cx + 3Dx^2 + 3Ex$$

Now, we group the terms by powers of ${ x }$:

$$8 x^4 + 3 x^3 - 115 x^2 - 39 x + 200 = (A + 3B)x^4 + 3Cx^3 + (20A + 30B + 3D)x^2 + (30C + 3E)x + 100A$$

By equating the coefficients of the corresponding powers of ${ x }$ on both sides, we obtain a system of linear equations:

1. Coefficient of ${ x^4 }$: ${ A + 3B = 8 }$
2. Coefficient of ${ x^3 }$: ${ 3C = 3 }$
3. Coefficient of ${ x^2 }$: ${ 20A + 30B + 3D = -115 }$
4. Coefficient of ${ x }$: ${ 30C + 3E = -39 }$
5. Constant term: ${ 100A = 200 }$

Solving this system of equations will give us the values of the constants ${ A, B, C, D, }$ and ${ E }$. From the fifth equation, we immediately find that ${ A = 2 }$. Substituting ${ A = 2 }$ into the first equation gives ${ 2 + 3B = 8 }$, so ${ 3B = 6 }$ and ${ B = 2 }$. From the second equation, we have ${ 3C = 3 }$, so ${ C = 1 }$. Substituting ${ A = 2 }$ and ${ B = 2 }$ into the third equation gives ${ 20(2) + 30(2) + 3D = -115 }$, so ${ 40 + 60 + 3D = -115 }$, which simplifies to ${ 3D = -215 }$, and ${ D = -\frac{215}{3} }$. Finally, substituting ${ C = 1 }$ into the fourth equation gives ${ 30(1) + 3E = -39 }$, so ${ 3E = -69 }$, and ${ E = -23 }$. Thus, we have found the values of all the constants.

Having determined the constants, we can now construct the partial fraction decomposition of the given rational function. We found that:

• ${ A = 2 }$
• ${ B = 2 }$
• ${ C = 1 }$
• ${ D = -\frac{215}{3} }$
• ${ E = -23 }$

Substituting these values into our partial fraction decomposition form, we get:

$$\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} = \frac{2}{3x} + \frac{2x + 1}{x^2 + 10} + \frac{-\frac{215}{3}x - 23}{(x^2 + 10)^2}$$

This is the partial fraction decomposition of the given rational function. It expresses the original complex rational function as a sum of simpler fractions, each of which is easier to handle in further mathematical operations, such as integration or differentiation. The process of finding these constants involved setting up a system of linear equations by equating coefficients, a standard technique in partial fraction decomposition. The accuracy of this decomposition can be verified by combining the fractions on the right-hand side back into a single fraction and checking if it matches the original rational function.

Now, let's revisit the original question: Which sum represents the partial fraction decomposition of ${ \frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} }$? We have already established the form of the decomposition as:

$$\frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} = \frac{A}{3x} + \frac{Bx + C}{x^2 + 10} + \frac{Dx + E}{(x^2 + 10)^2}$$

Considering the options provided in the original question, we can see that option B, ${ \frac{A}{3 x} + \frac{B x+C}{x^2+10} }$, is the closest to the first two terms of our derived decomposition. However, it is incomplete as it does not account for the repeated irreducible quadratic factor ${ (x^2 + 10)^2 }$ in the denominator. Option A, ${ \frac{A}{3 x} + \frac{B}{x^2+10} }$, is incorrect because it does not include the linear term in the numerator of the fraction associated with the irreducible quadratic factor.

To accurately represent the partial fraction decomposition, we need a sum that includes terms for each factor in the denominator, including the repeated quadratic factor. The correct form must account for ${ \frac{A}{3x} }$ for the linear factor ${ 3x }$, ${ \frac{Bx + C}{x^2 + 10} }$ for the quadratic factor ${ x^2 + 10 }$, and ${ \frac{Dx + E}{(x^2 + 10)^2} }$ for the repeated quadratic factor ${ (x^2 + 10)^2 }$. Therefore, the correct sum must have the form:

$$\frac{A}{3x} + \frac{Bx + C}{x^2 + 10} + \frac{Dx + E}{(x^2 + 10)^2}$$

Without the specific options listed in the original question, we can definitively say that the correct sum must include these three terms to fully represent the partial fraction decomposition of the given rational function.

In conclusion, the partial fraction decomposition of the rational function ${ \frac{8 x^4+3 x^3-115 x^2-39 x+200}{3 x\left(x^2+10\right)^2} }$ requires careful consideration of the factors in the denominator. The presence of a linear factor and a repeated irreducible quadratic factor dictates the form of the decomposition. We have demonstrated that the correct sum representing the partial fraction decomposition must include terms corresponding to each of these factors:

$$\frac{A}{3x} + \frac{Bx + C}{x^2 + 10} + \frac{Dx + E}{(x^2 + 10)^2}$$

This decomposition allows us to rewrite the complex rational function as a sum of simpler fractions, making it easier to integrate, differentiate, or manipulate algebraically. The process involves determining the constants ${ A, B, C, D, }$ and ${ E }$ by setting up and solving a system of linear equations. This technique is a fundamental tool in calculus and algebra, and a thorough understanding of it is essential for solving a wide range of mathematical problems. By breaking down complex rational functions into simpler components, partial fraction decomposition provides a powerful method for simplifying mathematical expressions and solving problems more efficiently. The principles and steps outlined in this article provide a solid foundation for mastering this important technique.