Simplifying Rational Expressions A Step By Step Guide

In mathematics, simplifying rational expressions is a fundamental skill. This guide will walk you through the process of simplifying two distinct rational expressions, providing a step-by-step approach and clear explanations. The goal is to reduce the expressions to their simplest forms, making them easier to work with in further calculations or algebraic manipulations. The first expression we will tackle is a rational expression involving quadratic polynomials, while the second involves fractions within fractions, requiring a slightly different approach.

Simplifying $\frac{6x^2 - 7x + 2}{4x^2 - 1}$

This section delves into the simplification of the rational expression $\frac{6x^2 - 7x + 2}{4x^2 - 1}$. The key to simplifying this expression lies in factoring both the numerator and the denominator. Factoring allows us to identify common factors that can be canceled out, leading to a simplified form. Let's break down the process step by step:

1. Factor the Numerator

The numerator is a quadratic expression: $6x^2 - 7x + 2$. To factor this, we look for two numbers that multiply to the product of the leading coefficient (6) and the constant term (2), which is 12, and add up to the middle coefficient (-7). These numbers are -3 and -4. We can then rewrite the middle term using these numbers:

$6x^2 - 7x + 2 = 6x^2 - 3x - 4x + 2$

Now, we factor by grouping:

$6x^2 - 3x - 4x + 2 = 3x(2x - 1) - 2(2x - 1)$ = $(3x - 2)(2x - 1)$

Therefore, the factored form of the numerator is $(3x - 2)(2x - 1)$. Factoring the numerator is a crucial step in simplifying rational expressions. It allows us to identify potential common factors with the denominator, which can then be canceled out to reduce the expression to its simplest form. The ability to factor quadratic expressions accurately is essential for this process.

2. Factor the Denominator

The denominator is a difference of squares: $4x^2 - 1$. This can be factored using the formula $a^2 - b^2 = (a + b)(a - b)$. In this case, $a = 2x$ and $b = 1$:

$4x^2 - 1 = (2x + 1)(2x - 1)$

Hence, the factored form of the denominator is $(2x + 1)(2x - 1)$. Recognizing and applying the difference of squares pattern is a valuable skill in simplifying algebraic expressions. It allows for quick and efficient factorization, which is particularly useful in this context.

3. Simplify the Expression

Now, we can rewrite the original expression with the factored forms of the numerator and denominator:

$\frac{6x^2 - 7x + 2}{4x^2 - 1} = \frac{(3x - 2)(2x - 1)}{(2x + 1)(2x - 1)}$

We can see that the factor $(2x - 1)$ appears in both the numerator and the denominator. We can cancel out this common factor, provided that $2x - 1 \neq 0$ (i.e., $x \neq \frac{1}{2}$):

$\frac{(3x - 2)(2x - 1)}{(2x + 1)(2x - 1)} = \frac{3x - 2}{2x + 1}$

Thus, the simplified form of the expression is $\frac{3x - 2}{2x + 1}$. This simplified form is easier to work with and provides a clearer understanding of the expression's behavior. The process of canceling common factors is the core of simplifying rational expressions.

4. State the Restrictions

It's important to state the restrictions on the variable $x$. These restrictions are values of $x$ that would make the original denominator equal to zero, which would make the expression undefined. From the factored denominator $(2x + 1)(2x - 1)$, we can see that the expression is undefined when $2x + 1 = 0$ or $2x - 1 = 0$. Solving these equations gives us:

$2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$

$2x - 1 = 0 \Rightarrow x = \frac{1}{2}$

Therefore, the restrictions are $x \neq -\frac{1}{2}$ and $x \neq \frac{1}{2}$. Stating the restrictions is crucial for a complete solution. It ensures that the simplified expression is equivalent to the original expression for all valid values of the variable. Ignoring restrictions can lead to incorrect interpretations or calculations.

Simplifying $\frac{\frac{4}{x} - \frac{3}{y}}{\frac{12}{xy}}$

This section focuses on simplifying a complex fraction: $\frac{\frac{4}{x} - \frac{3}{y}}{\frac{12}{xy}}$. Complex fractions, which involve fractions within fractions, often appear daunting but can be simplified systematically. The key to simplifying complex fractions is to eliminate the inner fractions. This can be achieved by finding a common denominator for the fractions in the numerator and then simplifying the resulting expression. Let's break down the process step by step:

1. Simplify the Numerator

The numerator is $\frac{4}{x} - \frac{3}{y}$. To subtract these fractions, we need a common denominator, which is $xy$. We rewrite each fraction with this denominator:

$\frac{4}{x} - \frac{3}{y} = \frac{4y}{xy} - \frac{3x}{xy}$

Now, we can subtract the fractions:

$\frac{4y}{xy} - \frac{3x}{xy} = \frac{4y - 3x}{xy}$

Thus, the simplified form of the numerator is $\frac{4y - 3x}{xy}$. Simplifying the numerator is a critical step in handling complex fractions. It consolidates the fractions into a single fraction, making the subsequent division step more straightforward.

2. Divide by the Denominator

The original expression can now be rewritten as:

$\frac{\frac{4y - 3x}{xy}}{\frac{12}{xy}}$

Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiply the numerator by the reciprocal of the denominator:

$\frac{\frac{4y - 3x}{xy}}{\frac{12}{xy}} = \frac{4y - 3x}{xy} \cdot \frac{xy}{12}$

3. Simplify the Expression

Now, we can simplify the expression by canceling out common factors. We see that $xy$ appears in both the numerator and the denominator, so we can cancel them out, provided that $xy \neq 0$ (i.e., $x \neq 0$ and $y \neq 0$):

$\frac{4y - 3x}{xy} \cdot \frac{xy}{12} = \frac{4y - 3x}{12}$

Therefore, the simplified form of the expression is $\frac{4y - 3x}{12}$. This simplified form is much easier to understand and work with compared to the original complex fraction. The process of converting division into multiplication by the reciprocal is a key technique in simplifying complex fractions.

4. State the Restrictions

It's important to state the restrictions on the variables $x$ and $y$. From the original expression, we can see that the denominators $x$, $y$, and $xy$ cannot be zero. Additionally, the denominator of the complex fraction, $\frac{12}{xy}$, cannot be zero, which implies $xy \neq 0$. Thus, the restrictions are $x \neq 0$ and $y \neq 0$.

Stating the restrictions is essential for a complete solution. It ensures that the simplified expression is equivalent to the original expression for all valid values of the variables. Failing to state restrictions can lead to errors in subsequent calculations or interpretations.

Simplifying rational expressions is a crucial skill in algebra. By mastering techniques like factoring, canceling common factors, and handling complex fractions, you can effectively reduce expressions to their simplest forms. Remember to always state the restrictions on the variables to ensure the equivalence of the original and simplified expressions. The ability to simplify rational expressions is fundamental for solving equations, working with functions, and tackling more advanced mathematical concepts.