Problem 94 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{2 x+1}{x^{4}-81}+\frac{2-x}{x^{4}-81}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{(x-3)(x^2 + 9)}. \)

Step 1: Recognize the Common Denominator

Both rational expressions have the same denominator, which is \( x^4 - 81 \). Since the denominators are the same, we can combine the numerators together directly over the common denominator.

Step 2: Combine the Numerators

Add the numerators: \( (2x+1) + (2-x) = 2x + 1 + 2 - x. \) Simplify the expression: \( 2x - x + 1 + 2 = x + 3. \)

Step 3: Simplify the Combined Expression

The expression becomes \( \frac{x+3}{x^4 - 81}. \) Now check if further simplification is possible.

Step 4: Factor the Denominator

Notice that \( x^4 - 81 \) is a difference of squares: \( x^4 - 81 = (x^2 - 9)(x^2 + 9). \) The term \( x^2 - 9 \) can be further factored as \( (x-3)(x+3). \) Thus, the complete factorization is \( (x-3)(x+3)(x^2 + 9). \)

Step 5: Simplify the Fraction by Cancelling Terms

The numerator \( x+3 \) is a factor in the denominator. Cancel \( x+3 \) from both the numerator and denominator: the expression simplifies to \( \frac{1}{(x-3)(x^2 + 9)}. \)

Key Concepts

Addition and Subtraction of Rational Expressions

When dealing with rational expressions, understanding how to add or subtract them is crucial. Rational expressions are fractions where both the numerator and the denominator are polynomials. To add or subtract these expressions, you need a common denominator, just like with numeric fractions.
In the given exercise, we have two rational expressions, \( \frac{2x+1}{x^4-81} \) and \( \frac{2-x}{x^4-81} \), which share the same denominator \( x^4-81 \). This simplifies our task since we can directly add the numerators together:

• Add: \( (2x+1) + (2-x) = 2x + 1 + 2 - x \).
• Simplify: \( 2x - x + 1 + 2 = x + 3 \).

Consequently, the combined expression is \( \frac{x+3}{x^4-81} \).
The process becomes a lot straightforward with a common denominator. Thus, finding and using a shared denominator helps keep the work tidy and manageable.

Factoring Polynomials

Factoring polynomials is a technique to break down expressions into products of simpler expressions. This skill is invaluable, particularly when simplifying rational expressions.
In the denominator \( x^4 - 81 \), you'll notice the form of a difference of squares:

• \( x^4 - 81 = (x^2)^2 - 9^2 \).

This identity lets you write it as \( (x^2 - 9)(x^2 + 9) \). But, there's more factorization to be done here:

• Notice: \( x^2 - 9 = (x-3)(x+3) \).

Therefore, we completely factor the denominator as \( (x-3)(x+3)(x^2 + 9) \). This step is crucial because it opens possibilities for simplifying the rational expression further.
The ability to factor polynomials swiftly makes solving and simplifying much smoother. Keep practicing this, and soon it will become second nature!

Simplifying Rational Expressions

Simplifying rational expressions often involves canceling terms after factorization. Unlike simplifying numbers, simplification in algebraic expressions requires keen inspection for common factors.
For the expression \( \frac {x+3}{(x-3)(x+3)(x^2+9)} \), notice that \( x+3 \) from the numerator is also a factor in the denominator. You can cancel these out:

• Result: \( \frac{1}{(x-3)(x^2+9)} \).

This process highlights why factorization is fundamental. It exposes potential cancellations and further reductions.
It's vital, however, to ensure you're canceling terms correctly. Canceling non-factors or making mistakes in factorization can lead to incorrect results. Always double-check your work to ensure each step is valid. By mastering these steps, simplifying rational expressions can become a manageable and even enjoyable task.