Problem 67

Question

simplify each complex rational expression. $$ \frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}} $$

Step-by-Step Solution

Verified

Answer

The simplified complex rational expression is $\frac{6}{7}$.

1Step 1: Factor the denominator

First, start by factoring the denominator expression $x^{2}-4$. This can be done because it is a difference of two squares. Therefore, $x^{2}-4$ can be factored into $(x-2)(x+2)$.

2Step 2: Simplify the numerator

The numerator has two parts: $\frac{3}{x-2}$ and $\frac{4}{x+2}$ which are subtracted. Multiply the two fractions by $(x-2)(x+2)$/$(x-2)(x+2)$ to unify their denominators. The numerator then simplifies to $3(x+2) - 4(x-2)$. When you simplify the expression, you get $6$.

3Step 3: Combine the numerator and the denominator

Combine the above simplified expressions for the numerator and the denominator to get $\frac{6}{(x-2)(x+2)}$.

4Step 4: Divide the fraction by the denominator's fraction

Replace the denominator of $\frac{7}{x^{2}-4}$ by its factored form $(x-2)(x+2)$ to get $\frac{6/(x-2)(x+2)}{7/(x-2)(x+2)}$. This simplifies to $\frac{6}{7}$ as the $(x-2)(x+2)$ in the numerator and the denominator cancel out.

Key Concepts

Factoring PolynomialsSimplifying FractionsDifference of SquaresAlgebraic Expressions

Factoring Polynomials

Polynomials are algebraic expressions that include terms made up of variables raised to whole number exponents, multiplied by coefficients. Factoring polynomials means breaking them down into simpler

expressions
that multiply together to get back to the original polynomial.

One of the key techniques for factoring is identifying patterns or special forms, such as the "difference of squares."
In our exercise, factoring the denominator was essential. The original quadratic expression was:

$x^2 - 4$.

This expression fits the pattern of a difference of squares, which can be expressed generally as

$a^2 - b^2 = (a - b)(a + b)$.

In our case, we can factor it to

$(x-2)(x+2)$.

Factoring makes it easier to cancel out common terms when simplifying expressions. This step sets the stage for further simplification later in the problem.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form so they are easier to work with.

Involves canceling out common factors in the numerator and the denominator.

A fraction is fully simplified when no further division is possible. In complex rational expressions, like in our problem, simplification often involves multiple steps. Let’s break down our numerator:

The expression was initially mixed with different denominators, $\frac{3}{x-2}$ and $\frac{4}{x+2}$.
To simplify it, a common denominator $(x-2)(x+2)$ was used.

Multiplying all terms by the common denominator allowed us to subtract correctly, converting it into a single fraction. Then, after simplifying, the numerator becomes a simple calculation:

$3(x+2) - 4(x-2)$.

Once simplified fully, this step will lead us to a fraction ready to be further reduced.

Difference of Squares

The difference of squares is a special algebraic identity useful in simplifying expressions.

It describes a scenario where two terms are squared and then subtracted.

The pattern follows the identity:

$a^2 - b^2 = (a+b)(a-b)$.

In math problems, recognizing this form allows quick factoring and simplification.In our original exercise, we exploited this concept.

The expression $x^2 - 4$ is indeed a difference of squares.

Here, where $a = x$ and $b = 2$, applying the formula results in the factors $(x-2)(x+2)$.

This factoring aids in simplifying complex expressions by providing terms that can "cancel out" in further simplification.

Algebraic Expressions

Algebraic expressions are combinations of different elements such as constants, variables, and mathematical operators. They are fundamental to algebra and help in expressing mathematical relationships. In the context of our exercise, it’s vital to manage algebraic expressions efficiently. Simplifying the complex rational fraction involved: