Problem 36 - Algebra and Trigonometry

Question

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{3 x^{2}+12 x-20}{x^{4}-8 x^{2}+16} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is a sum of fractions with terms from the factorized denominator.

Step 1: Factor the Denominator

The first step in finding the partial fraction decomposition is to factor the denominator polynomial fully. The denominator \( x^4 - 8x^2 + 16 \) can be rewritten as \( (x^2 - 4)^2 \) using the difference of squares: \( (x^2 - 4) = (x - 2)(x + 2) \). Thus, \( x^4 - 8x^2 + 16 \) factors to \( [(x - 2)(x + 2)]^2 \).

Step 2: Decompose into Partial Fractions

After factorizing the denominator, express the rational function as a sum of partial fractions. For the factorization \( [(x - 2)(x + 2)]^2 \), the decomposition is: \[ \frac{3x^2 + 12x - 20}{[(x-2)(x+2)]^2} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x+2} + \frac{D}{(x+2)^2} \]where \( A, B, C, \) and \( D \) are constants to be determined.

Step 3: Clear the Denominator

Multiply both sides of the equation by the denominator \( [(x-2)(x+2)]^2 \) to get rid of it. This gives you:\[ 3x^2 + 12x - 20 = A(x+2)^2 + B(x+2)(x-2) + C(x-2)^2 + D(x-2)(x+2) \].

Step 4: Expand and Combine Like Terms

Expand each term on the right-hand side and combine like terms, then equate the coefficients of corresponding powers of \( x \) on both sides of the equation to get a system of equations.

Step 5: Solve the System of Equations

Determine values for \( A, B, C, \) and \( D \) by solving the system of equations derived from equating coefficients. This will typically involve substitution or matrix methods involving the coefficients from both sides of the equation after expansion.

Step 6: Write the Partial Fraction Decomposition

Substitute the obtained values of \( A, B, C, \) and \( D \) back into the partial fraction decomposition to express the original rational function.

Key Concepts

Rational Function

A rational function is a type of function that involves the ratio of two polynomials. The general form is \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. These functions are very versatile, with properties that allow for simplification, solving, and decomposition.
In this exercise, the rational function is given by \( \frac{3x^2 + 12x - 20}{x^4 - 8x^2 + 16} \). The numerator and the denominator are polynomial expressions.
The goal is to simplify this expression by decomposing it into simpler parts, called partial fraction decomposition, which involves splitting the rational function into a sum of simpler rational functions.

Factorization

Factorization is a key step in simplifying rational functions. This involves rewriting a polynomial as a product of its simpler building blocks, called factors.
You can think of factorizing as the opposite of expanding. We are going to break down complex expressions into simpler ones. This exercise requires factorizing the denominator of the rational function \( x^4 - 8x^2 + 16 \).
By recognizing patterns like the difference of squares, you can rewrite it as \((x^2 - 4)(x^2 - 4)\), which is \([(x-2)(x+2)]^2\). This will allow us to express the function in a way that separates it into simpler rational expressions, paving the way for partial fraction decomposition.

System of Equations

A system of equations is a set of equations with multiple variables that you solve together. Solving these is crucial in partial fraction decomposition to find unknown constants in your expression.
After factorizing, you form a new equation by expressing your rational function as a sum of simpler fractions, which include unknown coefficients \(A\), \(B\), \(C\), and \(D\).
Multiplying out and expanding the terms allows you to match like terms, setting up a system of equations based on these coefficients. These systems can be solved using several methods such as substitution or matrix techniques, helping you find the precise values of the unknowns so that each part of the decomposition lines up correctly with the original rational function.

Polynomial Expansion

Polynomial expansion involves distributing and combining terms in expressions like \((x+2)^2\) or \( (x-2)^2 \).
This step is important in partial fraction decomposition to simplify the equation after we have matched the structure to potential partial fractions.
Expanding each term forcibly aligns like terms, allowing us to equate different powers of \(x\). This makes solving the system of equations easier, as you can directly compare coefficients from the expanded expression to those in the original rational function.
Expanding properly ensures that you have all the terms you need to form a complete and correct set of equations, leading to a successful decomposition.