Problem 68 - Algebra and Trigonometry

Question

simplify each complex rational expression. $$ \frac{\frac{x}{x-2}+1}{\frac{3}{x^{2}-4}+1} $$

Step-by-Step Solution

Answer

The simplified complex rational expression is \(2x^{2}-8/(x+1)\)

Step 1: Simplify Inner Fractions

The first thing to do is simplify the inner fractions inside the complex fraction. In this case, that means adding the fractions in the numerator and the denominator separately. The numerator simplifies to \(\frac{x}{x-2} + 1 = \frac{x+(x-2)}{x-2} = \frac{2x-2}{x-2}\) and the denominator simplifies to \(\frac{3}{x^{2}-4} + 1 = \frac{3}{(x-2)(x+2)} + \frac{(x-2)(x+2)}{(x-2)(x+2)} = \frac{3+x^{2}-4}{(x-2)(x+2)} = \frac{x^{2}-1}{(x-2)(x+2)}\).

Step 2: Simplify Complex Fraction

Next, simplify the complex fraction by dividing the numerator by the denominator. It turns into \(\frac{\frac{2x-2}{x-2}}{\frac{x^{2}-1}{(x-2)(x+2)}}\). Multiplication by the reciprocal of the divisor simplifies this to \(\frac{(2x-2)(x-2)(x+2)}{x^{2}-1}\).

Step 3: Factor and Simplify Further

Further simplify by factoring terms both in the numerator and denominator. The numerator is \(2(x-1)(x-2)(x+2)\) and the denominator is \((x+1)(x-1)\). Now, divide out the common factor of \((x-1)\) to simplify to \(2(x-2)(x+2)/(x+1)\).

Step 4: Finally Simplify Expression

Finally, open the brackets in the numerator and denominator, we get an expression of \(2(x^{2}-4)/(x+1) = 2x^{2}-8/(x+1)\). This is the simplified complex rational expression.

Key Concepts

Simplifying Fractions

Simplifying fractions is one of the most important concepts in algebra.It involves reducing the fraction to its simplest form, where the numerator and denominator have no common factors besides 1.In complex fractions, this process also involves simplifying fractions within a fraction.
Let's say you have a fraction like \( \frac{\frac{x}{x-2} + 1}{\frac{3}{x^2-4} + 1} \).You first need to reduce the smaller fractions in the numerator and the denominator.

• In the numerator, combine \( \frac{x}{x-2} \) with 1 by rewriting 1 as a fraction: \( \frac{x-2}{x-2} \).
• The denominator involves combining \( \frac{3}{x^2-4} \) with 1, by using the common denominator \((x-2)(x+2)\).

Understanding these small steps is key to simplifying complex fractions efficiently.

Factoring Expressions

Factoring expressions primarily involves breaking down a polynomial into a product of its factors.This is often a crucial step in both simplifying and solving expressions.
In our exercise, when you are left with \( \frac{2(x-1)(x-2)(x+2)}{(x+1)(x-1)} \), notice how factoring helps break down the numerator and denominator.You look for expressions in the form of \((a-b)(a+b)\) or \(x(a)\) patterns, which make them easy to factor and simplify.

• For the numerator: You identify common terms to factor, such as extracting a factor from \(2x-2 = 2(x-1)\).
• The denominator \(x^2 - 1\) is simplified to \((x-1)(x+1)\) using the difference of squares.

Factoring is a vital tool to simplify expressions to their easiest form.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are common in algebra problems and often need simplifying or factorizing.
In more complex problems, like our exercise, handling these expressions often involves factoring and simplifying to make them manageable. Rationalizing involves making them easy to handle by getting rid of complex, fractional numerators or denominators.

• Manipulating rational expressions requires common denominators for addition or subtraction.
• When multiplying or dividing, factorization is key to revealing possible simplifications.

By understanding these processes, you ensure your work with rational expressions is accurate and efficient.

Algebraic Fractions

Algebraic fractions, often seen as rational expressions in algebra, are fractions that involve variables.They follow the same rules as numeric fractions when it comes to addition, subtraction, multiplication, and division.
In our problem, we look at fractions such as \(\frac{x}{x-2} \) and \(\frac{3}{x^2-4} \) which require careful manipulation and simplification.The overall goal is to reach the simplest form where none of the algebraic components can be reduced further.

• First, always aim to simplify each part of the fraction before tackling the fraction as a whole.
• Then, apply concepts like factoring and common denominators to resolve complex operations.

Mastering algebraic fractions is crucial because they often appear in higher-level algebra and calculus problems.