Problem 27 - Precalculus

Question

Find the partial-fraction decomposition for each rational function. $$\frac{5 x^{2}+28 x-6}{(x+4)\left(x^{2}+3\right)}$$

Step-by-Step Solution

Answer

\(\frac{2}{x+4} + \frac{3x+16}{x^2+3}\).

Step 1: Identify the Partial Fraction Form

The given rational function is \( \frac{5x^2+28x-6}{(x+4)(x^2+3)} \). We identify the form for partial fraction decomposition based on the factors in the denominator. Since \(x+4\) is a linear factor and \(x^2+3\) is an irreducible quadratic factor, the partial fraction form is: \[ \frac{A}{x+4} + \frac{Bx+C}{x^2+3}. \]

Step 2: Multiply Through by the Denominator

Multiply the entire equation by the denominator \((x+4)(x^2+3)\) to clear the denominators: \[ 5x^2 + 28x - 6 = A(x^2 + 3) + (Bx + C)(x + 4). \] This eliminates the fractions and gives us a polynomial equation.

Step 3: Expand and Combine Like Terms

Expand the right-side terms: \[ A(x^2 + 3) = Ax^2 + 3A \] and \[ (Bx + C)(x + 4) = Bx^2 + 4Bx + Cx + 4C. \] Combine like terms: \[ (A + B)x^2 + (4B + C)x + (3A + 4C). \]

Step 4: Set Up a System of Equations

Align the coefficients from both sides for \(x^2\), \(x\), and the constant terms: 1. \( A + B = 5 \)2. \( 4B + C = 28 \)3. \( 3A + 4C = -6 \)

Step 5: Solve the System of Equations

First, solve for \(A\) and \(B\) from the first equation: - Substitute \(A = 5 - B\) into the third equation. - Plug into \(3(5 - B) + 4C = -6\):- Expand and simplify: \(15 - 3B + 4C = -6\)- Rearrange to: \(-3B + 4C = -21\) (Equation 4)From Equation 2: - \(4B + C = 28\)- Solve Equation 4 and Equation 2 simultaneously using substitution or elimination to find \(B\) and \(C\).Solving gives:- \(B = 3\) and \(C = 16\)Substitute \(B\) back to find \(A\):- \(A = 5 - B = 5 - 3 = 2\).

Step 6: Write the Partial Fraction Decomposition

Substitute \(A\), \(B\), and \(C\) back into the partial fraction form: \[ \frac{2}{x+4} + \frac{3x+16}{x^2+3}. \] This is the partial fraction decomposition of the given rational function.

Key Concepts

Rational Functions

Rational functions are expressions that represent the division of two polynomials. They take the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not equal to zero. The nature of the numerator and the denominator is essential when dealing with rational functions, especially in partial fraction decomposition.
Understanding rational functions involves identifying their characteristics, such as intercepts, asymptotes, and the end behavior as the input variable approaches infinity or negative infinity. In this context, the denominator's factors, such as \(x+4\) and \(x^2+3\), are crucial since they determine how the function behaves and how it can be decomposed.
Rational functions are prevalent in various real-world models and in stages of calculus and higher algebra, where they are used to solve complex integrals by breaking them into simpler parts through decomposition.

Systems of Equations

Systems of equations come into play when solving partial fraction decomposition problems. This happens because we often need to equate and solve several expressions simultaneously. In the given solution, we end up with a system of equations after aligning the coefficients of corresponding terms from the expanded expressions.
When faced with multiple unknowns, like \(A\), \(B\), and \(C\), we create a system of equations based on the powers of \(x\) and constant terms. This allows us to solve for the unknown coefficients effectively:

• First: \( A + B = 5 \)
• Second: \( 4B + C = 28 \)
• Third: \( 3A + 4C = -6 \)

These can be solved using various methods, like substitution or elimination, depending on what is most efficient for the particular problem. This approach often simplifies the solution process in algebraic contexts.

Polynomial Long Division

Polynomial long division is a method used to simplify rational functions when the degree of the numerator is equal to or exceeds that of the denominator. In this process, much like standard number division, dividends (numerator) are divided by divisors (denominator) repeatedly, generating a series of quotients and remainders.
While this specific exercise doesn't require polynomial long division, understanding this concept can be crucial in partial fraction decomposition when reductions of polynomials are necessary to simplify a function before decomposition. Knowing how to perform polynomial long division effectively can simplify complex rational expressions and aid in setting the stage for decomposition into simpler fractions.

Algebraic Fractions

Algebraic fractions refer to fractions that have polynomials in either the numerator, the denominator, or both. Simplification and manipulation of these fractions are vital skills in algebra and calculus. These fractions grow particularly complex when interacting with partial fraction decomposition.
In partial fraction decomposition, algebraic fractions are broken down into more manageable components. This simplifies the process of integration or other algebraic manipulations. For example, breaking the complex fraction \( \frac{5x^2+28x-6}{(x+4)(x^2+3)} \) into \( \frac{2}{x+4} + \frac{3x+16}{x^2+3} \) is beneficial because it provides simpler fractions that can be managed more easily in various mathematical procedures.
Mastering algebraic fractions involves becoming proficient in operations like addition, subtraction, and factor manipulation, which form the foundation for managing more complicated algebraic expressions.