Problem 59 - College Algebra

Question

For the following exercises, 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

Combine the fractions and solve for coefficients in the partial fractions decomposition.

Step 1: Factorize Each Quadratic in the Denominator

First, we identify the quadratics in the denominators: \(x^2 - 16\), \(x^2 + 6x + 8\), and \(x^2 - 4x\). These can be factorized as follows:- \(x^2 - 16 = (x-4)(x+4)\)- \(x^2 + 6x + 8 = (x+2)(x+4)\)- \(x^2 - 4x = x(x-4)\)

Step 2: Find the Common Denominator

The least common denominator (LCD) must include all factors from each fraction's denominator. Thus, the LCD is \(x(x-4)(x+4)(x+2)\).

Step 3: Rewrite Each Fraction with the LCD

Rewrite each term so that they all have the denominator \(x(x-4)(x+4)(x+2)\).- For \(\frac{2x}{(x-4)(x+4)}\), multiply numerator and denominator by \(x(x+2)\).- For \(\frac{1-2x}{(x+2)(x+4)}\), multiply by \(x(x-4)\).- For \(\frac{x-5}{x(x-4)}\), multiply by \((x+4)\).

Step 4: Combine the Fractions into a Single Fraction

Combine the fractions under the common denominator:\[\frac{2x \, x \, (x+2)}{x(x-4)(x+4)(x+2)} - \frac{(1-2x) \, x \, (x-4)}{x(x-4)(x+4)(x+2)} - \frac{(x-5) \, (x+4)}{x(x-4)(x+4)(x+2)}.\]

Step 5: Simplify the Numerators

Expand and combine the numerators:- For the first term: \(2x^3 + 4x^2\)- For the second term: \(-2x^3 + (8-4x)x\)- For the third term: \(x^2 - x - 20\)Combine these to get the simplified numerator.

Step 6: Resolve into Partial Fractions

Set up the expression like this: \(\frac{A}{x} + \frac{B}{x-4} + \frac{C}{x+4} + \frac{D}{x+2}\) for the denominator \(x(x-4)(x+4)(x+2)\).Find \(A\), \(B\), \(C\), and \(D\) by equating coefficients of like terms in the numerator.

Key Concepts

Factoring Quadratics

Factoring quadratics is an important first step when working with partial fraction decomposition. It involves rewriting a quadratic expression, such as \( x^2 - 16 \), into a product of linear factors. This makes it easier to simplify or manipulate the expression later.

Consider the quadratic \( x^2 - 16 \). This is a difference of squares, which can be factored into \((x - 4)(x + 4)\).

Other quadratics, like \( x^2 + 6x + 8 \), can be factored by finding two numbers that:

• multiply to the constant term (here, 8)
• add up to the linear coefficient (here, 6)

In this example, 2 and 4 satisfy both conditions, allowing the factorization \((x + 2)(x + 4)\).

Knowing how to factorize quickly and accurately is a cornerstone of success in algebra.

Least Common Denominator

Finding the Least Common Denominator (LCD) is crucial when combining fractions with different denominators. The LCD is the smallest expression that includes all factors from each fraction's denominator.

To determine the LCD for denominators such as \(x^2 - 16\), \(x^2 + 6x + 8\), and \(x^2 - 4x\), factor each quadratic first. In this case:

• \( x^2 - 16 = (x-4)(x+4) \)
• \( x^2 + 6x + 8 = (x+2)(x+4) \)
• \( x^2 - 4x = x(x-4) \)

The LCD must then include each distinct factor. For these denominators, the LCD is \(x(x-4)(x+4)(x+2)\).

Having a common denominator simplifies addition or subtraction of fractions, allowing them to be easily combined into a single expression.

Combining Fractions

Once a common denominator is identified, fractions can be combined into a single fraction. Each fraction must be rewritten over a common denominator, ensuring each term has the same base for comparison.

Take the problem example:

• The first term \( \frac{2x}{(x-4)(x+4)} \) needs to be multiplied by \(x(x+2)\), so that it becomes \( \frac{2x \cdot x \cdot (x+2)}{x(x-4)(x+4)(x+2)} \).
• The second term \( \frac{1-2x}{(x+2)(x+4)} \) is multiplied by \(x(x-4)\).
• For \( \frac{x-5}{x(x-4)} \), multiply by \((x+4)\).

These steps allow conversion of each fraction to a standard form, facilitating easy addition or subtraction. Once all fractions share a common denominator, they can be combined into a single fraction, making complex expressions much more manageable.

Simplifying Expressions

After combining fractions, the next step is to simplify the expression. Start by expanding each numerator and then combining like terms.

Let's examine each fraction after aligning under the common denominator. Consider:

• First term expansion: \(2x^3 + 4x^2\).
• Second term: \(-2x^3 + (8-4x)x\).
• Third term: \(x^2 - x - 20\).

Combine these numerators to achieve simplification.
This can often involve reorganizing terms and conducting basic arithmetic to condense the expression to its simplest form.

Once simplified, the expression may be easier to analyze or factor again, depending on the required mathematics operations. Understanding how to simplify effectively is pivotal in solving algebraic problems precisely and efficiently.