Simplify Algebraic Fractions Step-by-Step Solution

This article provides a detailed walkthrough on how to simplify the given algebraic expression. We will break down each step, ensuring clarity and understanding of the process involved in simplifying rational expressions. This guide is designed for students and anyone looking to enhance their algebra skills.

Problem Statement

We are given the following expression to simplify:

$\frac{x-1}{x^2-x-12}+\frac{3-x}{x^2-3 x-4}$
$\frac{2 x^2-10}{(x+3)(x-4)(x+1)}, x \neq-4,-1,3$

Our goal is to simplify this complex fraction into its simplest form. This involves factoring, finding common denominators, combining fractions, and reducing the expression.

Step 1: Factor the Denominators

The first step in simplifying rational expressions is to factor the denominators. This will help us identify common factors and find the least common denominator (LCD).

Factoring $x^2 - x - 12$

We are looking for two numbers that multiply to -12 and add to -1. These numbers are -4 and 3. Thus, we can factor the quadratic as:

$x^2 - x - 12 = (x - 4)(x + 3)$

Factoring $x^2 - 3x - 4$

Similarly, we need two numbers that multiply to -4 and add to -3. These numbers are -4 and 1. Therefore, the quadratic factors as:

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

Now, our expression looks like this:

$\frac{x-1}{(x-4)(x+3)}+\frac{3-x}{(x-4)(x+1)}$
$\frac{2 x^2-10}{(x+3)(x-4)(x+1)}, x \neq-4,-1,3$

Step 2: Find the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest multiple that all denominators share. In this case, the denominators are $(x-4)(x+3)$ and $(x-4)(x+1)$. The LCD will include all unique factors from both denominators.

The unique factors are $(x-4)$, $(x+3)$, and $(x+1)$. Thus, the LCD is:

LCD = $(x-4)(x+3)(x+1)$

Step 3: Rewrite Fractions with the LCD

Now, we need to rewrite each fraction with the LCD as the denominator. This involves multiplying the numerator and denominator of each fraction by the factors needed to obtain the LCD.

First Fraction

The first fraction is $\frac{x-1}{(x-4)(x+3)}$. To get the LCD, we need to multiply the denominator by $(x+1)$. Therefore, we multiply both the numerator and the denominator by $(x+1)$:

$\frac{x-1}{(x-4)(x+3)} \cdot \frac{x+1}{x+1} = \frac{(x-1)(x+1)}{(x-4)(x+3)(x+1)}$

Expanding the numerator, we get:

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

Second Fraction

The second fraction is $\frac{3-x}{(x-4)(x+1)}$. To get the LCD, we need to multiply the denominator by $(x+3)$. Therefore, we multiply both the numerator and the denominator by $(x+3)$:

$\frac{3-x}{(x-4)(x+1)} \cdot \frac{x+3}{x+3} = \frac{(3-x)(x+3)}{(x-4)(x+3)(x+1)}$

Expanding the numerator, we get:

$\frac{3x + 9 - x^2 - 3x}{(x-4)(x+3)(x+1)} = \frac{9 - x^2}{(x-4)(x+3)(x+1)}$

Now, our expression looks like this:

$\frac{x^2 - 1}{(x-4)(x+3)(x+1)} + \frac{9 - x^2}{(x-4)(x+3)(x+1)}$
$\frac{2 x^2-10}{(x+3)(x-4)(x+1)}, x \neq-4,-1,3$

Step 4: Combine the Fractions

Since both fractions now have the same denominator, we can combine them by adding the numerators:

$\frac{x^2 - 1 + 9 - x^2}{(x-4)(x+3)(x+1)}$

Simplifying the numerator, we get:

$\frac{8}{(x-4)(x+3)(x+1)}$

So, the simplified expression is:

$\frac{8}{(x-4)(x+3)(x+1)}$
$\frac{2 x^2-10}{(x+3)(x-4)(x+1)}, x \neq-4,-1,3$

Step 5: Consider the Given Denominator

The problem provides a denominator $\frac{2 x^2-10}{(x+3)(x-4)(x+1)}$. We should compare this with our current simplified fraction to see if further simplification is needed.

Analyzing the Provided Denominator

The provided denominator is:

$\frac{2 x^2-10}{(x+3)(x-4)(x+1)}$

We can factor out a 2 from the numerator:

$\frac{2(x^2 - 5)}{(x+3)(x-4)(x+1)}$

This doesn't seem to simplify further with our current numerator, which is just 8. Therefore, we should rewrite our expression to see if it fits better in the context of the original problem.

Step 6: Restate the Original Problem

Let's restate the original problem to make sure we're on the right track:

Simplify:

$\frac{x-1}{x^2-x-12}+\frac{3-x}{x^2-3 x-4} - \frac{2 x^2-10}{(x+3)(x-4)(x+1)}, x \neq-4,-1,3$

We missed the subtraction part in the original simplification. Let's correct this.

Step 7: Incorporate the Subtraction

Now we need to subtract the third fraction from the sum of the first two:

$\frac{8}{(x-4)(x+3)(x+1)} - \frac{2(x^2 - 5)}{(x-4)(x+3)(x+1)}$

Since the denominators are the same, we can combine the numerators:

$\frac{8 - 2(x^2 - 5)}{(x-4)(x+3)(x+1)}$

Simplify the Numerator

Expand and simplify the numerator:

$8 - 2(x^2 - 5) = 8 - 2x^2 + 10 = 18 - 2x^2$

So, the expression becomes:

$\frac{18 - 2x^2}{(x-4)(x+3)(x+1)}$

Factor Out Common Factors

We can factor out a -2 from the numerator:

$\frac{-2(x^2 - 9)}{(x-4)(x+3)(x+1)}$

Now, notice that $x^2 - 9$ is a difference of squares, which can be factored as $(x - 3)(x + 3)$:

$\frac{-2(x - 3)(x + 3)}{(x-4)(x+3)(x+1)}$

Step 8: Cancel Common Factors

We can cancel the common factor of $(x+3)$ from the numerator and denominator:

$\frac{-2(x - 3)}{(x-4)(x+1)}$

So, the final simplified expression is:

$\frac{-2(x - 3)}{(x-4)(x+1)}$

Final Answer

The simplified form of the given expression is:

$\frac{-2(x - 3)}{(x-4)(x+1)}$

This comprehensive guide walked through the process of simplifying complex algebraic fractions. Each step was detailed to ensure clarity and understanding. Remember, the key steps include factoring, finding the LCD, rewriting fractions with the LCD, combining fractions, and canceling common factors. By following these steps, you can simplify even the most complex rational expressions. Understanding these steps is crucial for simplifying algebraic fractions, as it allows for a systematic approach to solving these types of problems. Factoring is often the first crucial step, enabling us to identify common terms and simplify expressions more efficiently. The least common denominator (LCD) acts as a bridge, allowing us to combine fractions that initially seem disparate. The ultimate goal is to reduce the expression to its simplest form, making it easier to work with in further calculations or applications.

When simplifying algebraic fractions, several key concepts come into play. First and foremost is the ability to factor polynomials. Factoring allows us to break down complex expressions into simpler terms, making it easier to identify common factors that can be canceled out. Understanding different factoring techniques, such as factoring by grouping, difference of squares, and quadratic factoring, is crucial. Combining fractions is another essential concept. This involves finding a common denominator so that fractions can be added or subtracted. The least common denominator (LCD) is the most efficient common denominator to use, as it simplifies the process and reduces the complexity of the resulting fraction. Finally, reducing fractions to their simplest form involves canceling out common factors from the numerator and denominator. This step is crucial for obtaining the most concise form of the expression and ensures that the fraction is in its simplest terms. A thorough understanding of these concepts enables you to tackle a wide range of algebraic simplification problems with confidence and precision. Remember that consistent practice and a solid grasp of these techniques are the keys to mastering algebraic simplifications.

When simplifying rational expressions, it’s easy to make mistakes if you're not careful. One common error is failing to factor expressions correctly. If you don't factor properly, you might miss common factors that could be canceled out, leading to an incorrect simplification. Another frequent mistake is not finding the least common denominator (LCD) when adding or subtracting fractions. Using a denominator that isn't the LCD can complicate the process and lead to larger, more complex fractions. Additionally, students often make errors when distributing negative signs across terms in the numerator when combining fractions. Always double-check to ensure you've applied the negative sign correctly to all terms. Canceling terms instead of factors is another significant mistake. Remember that you can only cancel factors that are common to both the numerator and the denominator, not individual terms. Finally, arithmetic errors in basic calculations can also derail your simplification efforts. To avoid these mistakes, always take your time, show your work, and double-check each step. Mastering the process of simplifying rational expressions requires attention to detail and a thorough understanding of the underlying algebraic principles. By being mindful of these common pitfalls, you can increase your accuracy and confidence in solving these types of problems.

To truly master the simplification of algebraic fractions, consistent practice is key. Working through a variety of problems will help you solidify your understanding of the concepts and techniques involved. Start with basic problems that involve simple factoring and combining of fractions, then gradually move on to more complex expressions that require advanced factoring techniques and multiple steps. Practice problems might include adding, subtracting, multiplying, and dividing rational expressions, as well as simplifying complex fractions. Make sure to focus on identifying common factors, finding the least common denominator (LCD), and reducing the expressions to their simplest forms. It’s also helpful to work through problems that involve different types of polynomials, such as quadratics, cubics, and higher-degree polynomials. Pay close attention to the steps involved in each problem and review your work to identify any errors or areas for improvement. By engaging in regular practice, you will not only enhance your skills in simplifying fractions but also develop a deeper understanding of algebraic principles. Remember, the more you practice, the more confident and proficient you will become in handling these types of problems. Practice is the cornerstone of algebraic proficiency and is crucial for long-term retention and success.