Partial Fraction Decomposition Explained Finding The Correct Form For Polynomial Quotients

When dealing with rational functions in mathematics, particularly in calculus and algebra, the process of partial fraction decomposition is a powerful technique. This technique allows us to break down complex rational expressions into simpler fractions, making them easier to integrate, differentiate, or analyze. In this comprehensive exploration, we will delve into the intricacies of partial fraction decomposition, focusing on polynomial quotients and remainders. Specifically, we will address the question: Which expression represents the correct form for the quotient and remainder, written as partial fractions, of $\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7}$? This problem exemplifies the core principles of partial fraction decomposition and provides a solid foundation for understanding more complex scenarios.

Partial Fraction Decomposition: A Detailed Overview

Partial fraction decomposition is a method used to express a rational function (a fraction where both the numerator and denominator are polynomials) as a sum of simpler fractions. This is particularly useful when the denominator of the original fraction can be factored. The technique is widely applied in integral calculus to simplify the integration of rational functions, as well as in various areas of engineering and physics.

The general idea behind partial fraction decomposition is to reverse the process of adding fractions. When we add fractions, we find a common denominator and combine the numerators. Partial fraction decomposition does the opposite: it starts with a single fraction and breaks it down into multiple fractions with simpler denominators.

The key steps in performing partial fraction decomposition involve:

1. Polynomial Long Division (if necessary): If the degree of the numerator is greater than or equal to the degree of the denominator, we first perform polynomial long division. This will result in a quotient and a remainder, where the degree of the remainder is less than the degree of the denominator.
2. Factor the Denominator: Factor the denominator of the rational function into linear and/or irreducible quadratic factors.
3. Set up the Partial Fractions: Based on the factors in the denominator, set up the partial fractions. For each linear factor $(ax + b)$, the partial fraction will have the form $\frac{A}{ax + b}$, where A is a constant. For each irreducible quadratic factor $(ax^2 + bx + c)$, the partial fraction will have the form $\frac{Ax + B}{ax^2 + bx + c}$, where A and B are constants.
4. Solve for the Constants: Multiply both sides of the equation by the original denominator to clear the fractions. Then, solve for the unknown constants (A, B, etc.) by substituting suitable values of x or by equating coefficients of like terms.
5. Write the Partial Fraction Decomposition: Substitute the values of the constants back into the partial fractions.

Applying Polynomial Long Division

In the given problem, we have the rational function $\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7}$. The degree of the numerator (3) is greater than the degree of the denominator (2), so we need to perform polynomial long division.

Performing the long division, we divide $6x^3 + 15x^2 + 20x + 36$ by $3x^2 + 7$. The steps are as follows:

1. Divide the leading term of the numerator ($6x^3$) by the leading term of the denominator ($3x^2$) to get $2x$. This is the first term of the quotient.
2. Multiply the entire denominator ($3x^2 + 7$) by $2x$ to get $6x^3 + 14x$.
3. Subtract this result from the numerator: $(6x^3 + 15x^2 + 20x + 36) - (6x^3 + 14x) = 15x^2 + 6x + 36$.
4. Divide the leading term of the new expression ($15x^2$) by the leading term of the denominator ($3x^2$) to get 5. This is the second term of the quotient.
5. Multiply the entire denominator ($3x^2 + 7$) by 5 to get $15x^2 + 35$.
6. Subtract this result from the remaining expression: $(15x^2 + 6x + 36) - (15x^2 + 35) = 6x + 1$.

Thus, the quotient is $2x + 5$ and the remainder is $6x + 1$. We can write the original rational function as:

$$\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7} = 2x + 5 + \frac{6x + 1}{3x^2 + 7}$$

Setting Up Partial Fractions

Now we focus on the remainder term, $\frac{6x + 1}{3x^2 + 7}$. The denominator $3x^2 + 7$ is an irreducible quadratic because it cannot be factored into linear factors with real coefficients (the discriminant $b^2 - 4ac = 0^2 - 4(3)(7) = -84$ is negative). Therefore, the partial fraction decomposition of this term will have the form:

$$\frac{6x + 1}{3x^2 + 7} = \frac{Ax + B}{3x^2 + 7}$$

Here, $Ax + B$ represents the most general form of a linear numerator over an irreducible quadratic denominator. This is because we need to account for the possibility of both a linear term ($Ax$) and a constant term ($B$) in the numerator.

Determining the Correct Form

Based on the polynomial long division and the setup of partial fractions, we can now identify the correct form for the quotient and remainder. The original rational function can be expressed as:

$$\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7} = 2x + 5 + \frac{Ax + B}{3x^2 + 7}$$

Comparing this expression with the given options, we can see that option B, $2x + 5 + \frac{Ax + B}{3x^2 + 7}$, matches our result. This is the correct form for the quotient and remainder, where $2x + 5$ is the quotient and $\frac{Ax + B}{3x^2 + 7}$ represents the remainder term.

Evaluating the Constants A and B

Although the question only asks for the correct form, let's go a step further and determine the values of A and B. We already have:

$$\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7} = 2x + 5 + \frac{6x + 1}{3x^2 + 7}$$

Comparing this with our partial fraction form:

$$2x + 5 + \frac{6x + 1}{3x^2 + 7} = 2x + 5 + \frac{Ax + B}{3x^2 + 7}$$

We can directly equate the numerators of the remainder terms:

$$6x + 1 = Ax + B$$

By comparing the coefficients of the like terms, we find:

• $A = 6$
• $B = 1$

Therefore, the complete partial fraction decomposition is:

$$\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7} = 2x + 5 + \frac{6x + 1}{3x^2 + 7}$$

Significance of Partial Fraction Decomposition

Partial fraction decomposition is not just a mathematical technique; it is a powerful tool with significant applications in various fields. Its primary use is in calculus, where it simplifies the integration of rational functions. Integrating a complex rational function can be challenging, but by decomposing it into simpler fractions, the integration process becomes much more manageable.

In engineering, partial fraction decomposition is used in circuit analysis, control systems, and signal processing. For example, in electrical engineering, it is used to analyze the transient response of circuits. In control systems, it helps in analyzing the stability and performance of feedback systems. In signal processing, it is used to design filters and analyze signals in the frequency domain.

Partial fraction decomposition also finds applications in physics, particularly in electromagnetism and quantum mechanics. It is used in solving differential equations and in analyzing various physical phenomena.

Common Mistakes and How to Avoid Them

When performing partial fraction decomposition, it is important to avoid common mistakes that can lead to incorrect results. Here are some common pitfalls and how to avoid them:

1. Forgetting Polynomial Long Division: If the degree of the numerator is greater than or equal to the degree of the denominator, polynomial long division must be performed first. Failing to do so will lead to an incorrect setup of the partial fractions.
2. Incorrectly Factoring the Denominator: The denominator must be factored correctly into linear and irreducible quadratic factors. An incorrect factorization will result in an incorrect partial fraction decomposition.
3. Incorrect Setup of Partial Fractions: The form of the partial fractions must match the factors in the denominator. For linear factors, use $\frac{A}{ax + b}$, and for irreducible quadratic factors, use $\frac{Ax + B}{ax^2 + bx + c}$.
4. Errors in Solving for Constants: Solving for the constants (A, B, etc.) requires careful algebraic manipulation. Common mistakes include errors in multiplying, adding, or subtracting terms.
5. Not Checking the Result: After completing the partial fraction decomposition, it is a good practice to check the result by adding the partial fractions back together to see if they equal the original rational function.

Advanced Techniques and Special Cases

While the basic principles of partial fraction decomposition are straightforward, there are advanced techniques and special cases that are worth exploring. One such case is when the denominator has repeated factors. For example, if the denominator has a factor of $(ax + b)^2$, the partial fraction decomposition will include two terms: $\frac{A}{ax + b}$ and $\frac{B}{(ax + b)^2}$.

Another advanced technique involves using complex numbers. In some cases, factoring the denominator over the complex numbers can simplify the partial fraction decomposition process. This is particularly useful when dealing with irreducible quadratic factors.

Conclusion

In summary, determining the correct form for the quotient and remainder in partial fraction decomposition involves several key steps: performing polynomial long division if necessary, factoring the denominator, setting up the partial fractions, solving for the constants, and writing the final decomposition. For the given problem, the correct form for the quotient and remainder of $\frac{6x^3 + 15x^2 + 20x + 36}{3x^2 + 7}$ is $2x + 5 + \frac{Ax + B}{3x^2 + 7}$.

Mastering partial fraction decomposition is essential for success in calculus and related fields. By understanding the underlying principles and practicing the techniques, students and professionals can confidently tackle complex rational functions and apply this powerful tool to solve a wide range of problems. This detailed exploration has provided a comprehensive understanding of partial fraction decomposition, its applications, and how to avoid common mistakes, ensuring a solid foundation for further studies in mathematics and engineering.

By understanding the process of partial fraction decomposition and its applications, you can confidently tackle similar problems and excel in your mathematical studies. Remember, practice is key to mastering this technique. Work through various examples, and don't hesitate to seek help when needed. With dedication and effort, you can become proficient in partial fraction decomposition and unlock its full potential.