Problem 59 - College Algebra

Question

Perform the operation and then find the partial fraction decomposition. \(\frac{2 x}{x^{2}-16}-\frac{1-2 x}{x^{2}+6 x+8}-\frac{x-5}{x^{2}-4 x}\)

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{A}{x} + \frac{B}{x-4} + \frac{C}{x+4} + \frac{D}{x+2}\), where A, B, C, and D are determined through solving.

Step 1: Identify the common denominator

The denominators of the fractions are \(x^2 - 16\), \(x^2 + 6x + 8\), and \(x^2 - 4x\), which we can factor. First, factor them separately: \(x^2 - 16 = (x-4)(x+4)\), \(x^2 + 6x + 8 = (x+2)(x+4)\), and \(x^2 - 4x = x(x-4)\). The common denominator will be \(x(x-4)(x+4)(x+2)\).

Step 2: Rewrite each fraction with the common denominator

Rewrite each fraction so that they have the common denominator \(x(x-4)(x+4)(x+2)\). Multiply the numerators accordingly: 1. \(\frac{2x}{(x-4)(x+4)} \rightarrow \frac{2x(x)(x+2)}{x(x-4)(x+4)(x+2)}\)2. \(\frac{1-2x}{(x+2)(x+4)} \rightarrow \frac{-(1-2x)x(x-4)}{x(x-4)(x+4)(x+2)}\)3. \(\frac{x-5}{x(x-4)} \rightarrow \frac{(x-5)(x+4)(x+2)}{x(x-4)(x+4)(x+2)}\).

Step 3: Combine the fractions

Now that all fractions have a common denominator, combine the numerators: \ \(2x(x)(x+2) - (1-2x)x(x-4) - (x-5)(x+4)(x+2)\). Simplify this expression by expanding each term.

Step 4: Simplify the equation

Calculate each part: - \(2x(x)(x+2) = 2x^3 + 4x^2\)- \( -(1-2x)x(x-4) = -x(x-4) + 2x^2(x-4) = -x^2 + 4x + 2x^3 - 8x^2\)- \((x-5)(x+4)(x+2) = (x^2 + 6x + 8)(x-5) = x^3 + 6x^2 + 8x - 5x^2 - 30x - 40 = x^3 + x^2 - 22x - 40\)Combine and simplify the whole numerator: \ \((2x^3 + 4x^2) - ( -x^2 + 4x + 2x^3 - 8x^2) - (x^3 + x^2 - 22x - 40)\).

Step 5: Simplify further

After simplification, the result is: \((2x^3 + 4x^2) - (-x^2 + 4x + 2x^3 - 8x^2) - (x^3 + x^2 - 22x - 40) = -x^3 + 11x^2 - 18x + 40\).

Step 6: Perform partial fraction decomposition

Now we need to decompose the expression \(\frac{-x^3 + 11x^2 - 18x + 40}{x(x-4)(x+4)(x+2)}\) into partial fractions. Assume the form:\[ \frac{A}{x} + \frac{B}{x-4} + \frac{C}{x+4} + \frac{D}{x+2} \]Multiply by the common denominator to solve for A, B, C, and D by comparing coefficients or substituting convenient values.

Step 7: Solve for unknown coefficients

Substitute different values for \(x\) to find the coefficients:- For \(x = 0\), solve for \(A\).- For \(x = 4\), solve for \(B\).- For \(x = -4\), solve for \(C\).- For \(x = -2\), solve for \(D\).Solving these gives the respective values of the coefficients.

Step 8: Write the partial fraction decomposition

Assuming the values from solving, the partial fraction decomposition is:\[ \frac{A}{x} + \frac{B}{x-4} + \frac{C}{x+4} + \frac{D}{x+2} \]Substitute back the calculated coefficients to get the final decomposed fractions.

Key Concepts

Rational Functions

Rational functions are a type of function in mathematics where one polynomial is divided by another. They can be expressed in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero. These functions are central in various mathematical analyses since they can model a broad range of real-world problems.

In this exercise, we work with rational functions involving multiple algebraic fractions. Each fraction has a polynomial expression in both the numerator and denominator. By performing operations on these functions, such as addition or subtraction, we need to find a common denominator. This combines the fractions into a single, simplified rational function.

Partial fraction decomposition is a valuable technique for breaking down complex fractions into simpler ones, making integration or solving polynomial equations easier. It essentially reverses the process of finding a common denominator and expresses the original function as a sum of simpler fractions.

Polynomial Equations

Polynomial equations are equations in which the expressions are made up of polynomials. A polynomial expression typically takes the form \( a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \), where the coefficients \( a_n, a_{n-1}, \, \ldots \, , a_1, a_0 \) are real numbers, and the powers of \( x \) are non-negative integers.

Solving polynomial equations often involves factoring them into simpler components, which can help simplify complex algebraic fractions. In our exercise, factoring the denominators of the original fractions allows us to find a common denominator across all terms. For instance:

• The polynomial \( x^2 - 16 \) is factored as \((x-4)(x+4)\).
• The polynomial \( x^2 + 6x + 8 \) becomes \((x+2)(x+4)\).
• The polynomial \( x^2 - 4x \) simplifies to \(x(x-4)\).

These factored forms make it easier to handle the fractions correctly, combining them into one rational function with a uniform denominator.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but instead of numbers, they consist of algebraic expressions. These fractions can be simplified, combined, and decomposed using various algebraic principles.

In dealing with algebraic fractions, especially when subtracting or adding them, it's crucial to have a common denominator. This allows the numerators to be manipulated while keeping the denominator consistent. Our example fraction involves three separate algebraic fractions, each with different denominators, but we combined them by finding a common denominator: \(x(x-4)(x+4)(x+2)\).

Once the equation is simplified, the numerator undergoes partial fraction decomposition. This technique splits a complex fraction into simpler components and involves solving for unknown coefficients. By setting up an equation in the form \( \frac{A}{x} + \frac{B}{x-4} + \frac{C}{x+4} + \frac{D}{x+2} \), you solve for \( A, B, C, \) and \( D \) using techniques like substitution, which helps in integrating or further simplifying the expression.