Simplifying Algebraic Expressions A Step-by-Step Guide

This comprehensive guide dives deep into the world of simplifying algebraic expressions, focusing on rational expressions. We will tackle two complex problems step-by-step, providing a clear understanding of the underlying principles and techniques involved. Whether you're a student grappling with algebra or a seasoned mathematician seeking a refresher, this guide offers valuable insights and practical solutions.

Problem 2.1 Simplifying ${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \div \frac{3y^2 + 3yz}{3}}$

Unveiling the Complexity of Rational Expressions

In the realm of algebra, rational expressions often appear daunting due to their intricate structures involving polynomials in both the numerator and denominator. However, with a systematic approach and a firm grasp of algebraic manipulations, simplifying these expressions becomes a manageable task. This section delves into the simplification of the given rational expression, highlighting the key steps and techniques involved. Our primary focus will be on factoring, canceling common factors, and performing operations on rational expressions. Understanding these concepts is not just crucial for solving this particular problem but also for tackling a wide array of algebraic challenges. The journey through this problem will serve as a cornerstone for building your proficiency in algebraic simplification.

Step 1 Factoring Expressions

The cornerstone of simplifying rational expressions lies in the ability to factor polynomials effectively. Factoring allows us to identify common factors between the numerator and denominator, which can then be canceled out, leading to a simplified expression. In our given expression, ${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \div \frac{3y^2 + 3yz}{3}}$, we begin by factoring each individual polynomial.

• Numerator of the first fraction: ${xy^2 - xz^2}$ can be factored by first extracting the common factor ${x}$, resulting in ${x(y^2 - z^2)}$. Recognizing the difference of squares pattern, ${y^2 - z^2}$ can be further factored into ${(y - z)(y + z)}$. Thus, the fully factored numerator is ${x(y - z)(y + z)}$.
• Denominator of the first fraction: ${y^2 - 2yz + z^2}$ is a perfect square trinomial, which factors into ${(y - z)^2}$ or ${(y - z)(y - z)}$. Recognizing this pattern is key to efficient simplification.
• Numerator of the second fraction: ${y - z}$ is already in its simplest form and cannot be factored further.
• Denominator of the second fraction: ${xy - xz}$ can be factored by extracting the common factor ${x}$, yielding ${x(y - z)}$.
• Numerator of the third fraction: ${3y^2 + 3yz}$ can be factored by extracting the common factor ${3y}$, resulting in ${3y(y + z)}$.
• Denominator of the third fraction: ${3}$ is a constant and is already in its simplest form.

Factoring is not merely a mechanical process; it requires a keen eye for patterns and relationships within the expressions. Mastery of factoring techniques, such as recognizing the difference of squares, perfect square trinomials, and extracting common factors, is paramount for simplifying rational expressions. This step lays the foundation for the subsequent steps, where we will utilize these factored forms to cancel common factors and simplify the expression.

Step 2 Rewriting Division as Multiplication

In the realm of mathematical operations, division can be elegantly transformed into multiplication by employing the concept of reciprocals. This transformation is particularly useful when dealing with rational expressions, as it allows us to consolidate the operations into a single multiplicative process. In our problem, we encounter division by the fraction ${\frac{3y^2 + 3yz}{3}}$. To convert this division into multiplication, we take the reciprocal of this fraction, which means swapping the numerator and denominator. The reciprocal of ${\frac{3y^2 + 3yz}{3}}$ is ${\frac{3}{3y^2 + 3yz}}$.

By rewriting division as multiplication, we effectively streamline the expression, making it easier to manipulate and simplify. The original expression, ${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \div \frac{3y^2 + 3yz}{3}}$, now transforms into ${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \times \frac{3}{3y^2 + 3yz}}$. This seemingly simple change sets the stage for the next crucial step: canceling common factors. The ability to seamlessly switch between division and multiplication, particularly with rational expressions, is a testament to a strong understanding of fundamental algebraic principles. This step not only simplifies the immediate problem but also reinforces a key concept in mathematical manipulation.

Step 3 Canceling Common Factors

After factoring and rewriting division as multiplication, the stage is set for the most visually satisfying part of simplifying rational expressions: canceling common factors. This process involves identifying factors that appear in both the numerator and the denominator of the expression and then eliminating them. The underlying principle is that any factor divided by itself equals 1, effectively removing it from the expression without changing its value. In our transformed expression, ${\frac{x(y - z)(y + z)}{(y - z)(y - z)} \times \frac{y-z}{x(y-z)} \times \frac{3}{3y(y + z)}}$, we can identify several common factors ready for cancellation.

• The factor ${(y - z)}$ appears in both the numerator and the denominator multiple times. We can cancel one ${(y - z)}$ from the first fraction's numerator with one from its denominator. We can also cancel the ${(y - z)}$ from the second fraction's numerator with another ${(y - z)}$ from the first fraction's denominator. This leaves us with no ${(y - z)}$ factors in the numerator and one ${(y - z)}$ in the denominator.
• The factor ${x}$ appears in the numerator of the first fraction and the denominator of the second fraction. These can be canceled out directly.
• The factor ${(y + z)}$ appears in the numerator of the first fraction and the denominator of the third fraction. These can also be canceled out.
• The constant ${3}$ appears in the numerator of the third fraction and the denominator of the third fraction. These can be canceled out as well.

After meticulously canceling these common factors, the expression is significantly simplified. The act of canceling common factors is not just about making the expression look simpler; it's about revealing the underlying structure and relationships within the expression. This step highlights the power of factoring and the elegance of mathematical simplification. By carefully identifying and canceling common factors, we move closer to the final, most simplified form of the expression.

Step 4 Multiplying Remaining Factors

Following the cancellation of common factors, the focus shifts to consolidating the remaining terms. This involves multiplying the factors that persist in the numerator and the denominator separately. The goal is to combine these remaining elements into a single, simplified fraction. In our case, after the extensive cancellation, we are left with a much leaner expression. Let's examine what remains in both the numerator and the denominator.

In the numerator, after canceling out ${x}$, ${(y - z)}$, and ${(y + z)}$, we are left with a 1. It's crucial to remember that when all factors are canceled, the remaining value is 1, not 0. This is because we are essentially dividing the factor by itself, which results in 1.

In the denominator, after canceling out ${(y - z)}$, ${x}$, ${(y + z)}$, and ${3}$, we are left with ${y}$. This means that the product of the remaining factors in the denominator is simply ${y}$.

Now, we combine the simplified numerator and denominator to form the final simplified expression. The numerator is 1, and the denominator is ${y}$. Therefore, the simplified expression is ${\frac{1}{y}}$. This step underscores the importance of paying close attention to detail and ensuring that all remaining factors are correctly accounted for. The process of multiplying remaining factors is a final consolidation, bringing together all the previous steps to arrive at the most concise form of the expression.

Step 5 State Restrictions

While simplifying algebraic expressions often leads to a more concise form, it's crucial to acknowledge the potential for introducing extraneous solutions or altering the domain of the expression. Restrictions are values that would make the original expression undefined, typically due to division by zero. Identifying and stating these restrictions is a critical step in ensuring the mathematical integrity of the simplification process.

To determine the restrictions, we must examine the denominators of the original expression and any intermediate expressions formed during the simplification process before any cancellations are made. This is because canceling a factor that could be zero effectively masks the restriction.

In our original expression, ${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \div \frac{3y^2 + 3yz}{3}}$, we have the following denominators to consider:

1. ${y^2 - 2yz + z^2}$ which factors to ${(y - z)^2}$
2. ${xy - xz}$ which factors to ${x(y - z)}$
3. ${3y^2 + 3yz}$ which factors to ${3y(y + z)}$

Setting each of these denominators to zero and solving for the variables will give us the restrictions:

1. ${(y - z)^2 = 0}$ implies ${y - z = 0}$, so ${y = z}$.
2. ${x(y - z) = 0}$ implies ${x = 0}$ or ${y - z = 0}$, so ${x = 0}$ or ${y = z}$.
3. ${3y(y + z) = 0}$ implies ${y = 0}$ or ${y + z = 0}$, so ${y = 0}$ or ${y = -z}$.

Combining these restrictions, we find that the original expression is undefined when ${x = 0}$, ${y = z}$, ${y = 0}$, or ${y = -z}$. These restrictions must be stated alongside the simplified expression to ensure its complete and accurate representation.

Stating restrictions is not just a formality; it's an essential part of mathematical rigor. It ensures that the simplified expression is only used for values where the original expression is defined, preserving the integrity of the mathematical statement. By explicitly stating these restrictions, we provide a complete and accurate solution to the simplification problem.

Solution for Problem 2.1

Thus, the simplified expression is:

${\frac{xy^2 - xz^2}{y^2 - 2yz + z^2} \times \frac{y-z}{xy-xz} \div \frac{3y^2 + 3yz}{3} = \frac{1}{y}}$

with the restrictions:

${x \ne 0, y \ne z, y \ne 0, y \ne -z}$

Problem 2.2 Simplifying ${\frac{2}{3x^2 - 4x + 1} - \frac{2}{x-1} + \frac{5}{6x^2 + x - 1}}$

Deconstructing Complex Rational Expressions

Rational expressions, characterized by polynomials in both the numerator and denominator, often present a challenge when it comes to simplification. The complexity arises from the need to combine fractions with different denominators, a process that demands meticulous attention to detail and a solid understanding of algebraic principles. This section focuses on dissecting the given expression, ${\frac{2}{3x^2 - 4x + 1} - \frac{2}{x-1} + \frac{5}{6x^2 + x - 1}}$, and employing a systematic approach to simplify it. Our journey will involve factoring quadratic expressions, finding the least common denominator (LCD), and combining fractions. Mastering these techniques is not only essential for this specific problem but also for navigating a wide spectrum of algebraic manipulations. The insights gained here will empower you to tackle intricate rational expressions with confidence and precision.

Step 1 Factoring Denominators

The initial step in simplifying a sum or difference of rational expressions is to factor the denominators. Factoring allows us to identify common factors and, more importantly, to determine the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators, and it is crucial for combining the fractions. In our expression, ${\frac{2}{3x^2 - 4x + 1} - \frac{2}{x-1} + \frac{5}{6x^2 + x - 1}}$, we have three denominators to factor.

• The first denominator, ${3x^2 - 4x + 1}$, is a quadratic trinomial. To factor it, we look for two numbers that multiply to ${3 \times 1 = 3}$ and add up to -4. These numbers are -3 and -1. We can then rewrite the middle term and factor by grouping: ${3x^2 - 3x - x + 1 = 3x(x - 1) - 1(x - 1) = (3x - 1)(x - 1)}$.
• The second denominator, ${x - 1}$, is already in its simplest form and cannot be factored further.
• The third denominator, ${6x^2 + x - 1}$, is another quadratic trinomial. We look for two numbers that multiply to ${6 \times -1 = -6}$ and add up to 1. These numbers are 3 and -2. Rewriting the middle term and factoring by grouping gives us: ${6x^2 + 3x - 2x - 1 = 3x(2x + 1) - 1(2x + 1) = (3x - 1)(2x + 1)}$.

Factoring these denominators is a critical step, as it lays the foundation for finding the LCD. The ability to factor quadratic trinomials efficiently is a key skill in algebra, and it is essential for simplifying rational expressions. By factoring the denominators, we gain a clear picture of the building blocks of each fraction, making it easier to combine them.

Step 2 Finding the Least Common Denominator (LCD)

After factoring the denominators, the next crucial step is to determine the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators. It serves as the common ground upon which we can combine the fractions. To find the LCD, we consider all the unique factors present in the denominators and take the highest power of each factor.

In our problem, the factored denominators are:

1. ${(3x - 1)(x - 1)}$
2. ${(x - 1)}$
3. ${(3x - 1)(2x + 1)}$

The unique factors present are ${(3x - 1)}$, ${(x - 1)}$, and ${(2x + 1)}$. Each of these factors appears with a power of 1. Therefore, the LCD is the product of these factors: ${(3x - 1)(x - 1)(2x + 1)}$.

The LCD is not merely a mathematical construct; it's the key to unlocking the simplification of rational expressions. By identifying the LCD, we create a common foundation that allows us to combine the fractions seamlessly. This step highlights the importance of understanding the relationships between the denominators and how they can be unified under a single expression. The LCD serves as a bridge, connecting the individual fractions into a cohesive whole.

Step 3 Rewriting Fractions with the LCD

Once the least common denominator (LCD) has been determined, the next step is to rewrite each fraction with the LCD as its new denominator. This process involves multiplying the numerator and denominator of each fraction by the factors that are missing from its original denominator compared to the LCD. This ensures that the value of the fraction remains unchanged while allowing us to combine them under a common denominator.

In our problem, the LCD is ${(3x - 1)(x - 1)(2x + 1)}$. Let's rewrite each fraction:

1. The first fraction, ${\frac{2}{(3x^2 - 4x + 1)}}$, can be written as ${\frac{2}{(3x - 1)(x - 1)}}$. To get the LCD, we need to multiply the numerator and denominator by ${(2x + 1)}$: ${\frac{2(2x + 1)}{(3x - 1)(x - 1)(2x + 1)}}$.
2. The second fraction, ${\frac{2}{(x - 1)}}$, needs to be multiplied by ${(3x - 1)(2x + 1)}$ in both the numerator and denominator: ${\frac{2(3x - 1)(2x + 1)}{(3x - 1)(x - 1)(2x + 1)}}$.
3. The third fraction, ${\frac{5}{(6x^2 + x - 1)}}$, can be written as ${\frac{5}{(3x - 1)(2x + 1)}}$. To get the LCD, we multiply the numerator and denominator by ${(x - 1)}$: ${\frac{5(x - 1)}{(3x - 1)(x - 1)(2x + 1)}}$.

Rewriting fractions with the LCD is a crucial step in preparing them for combination. It ensures that each fraction has the same denominator, allowing us to perform addition and subtraction operations on the numerators. This process highlights the importance of maintaining the value of the fraction while transforming its form. By carefully multiplying by the missing factors, we create a common foundation for combining the fractions, paving the way for simplification.

Step 4 Combining Numerators

With all the fractions now sharing the least common denominator (LCD), the next step is to combine the numerators. This involves performing the indicated operations (addition and subtraction) on the numerators while keeping the common denominator. The key here is to distribute any multiplication and then combine like terms in the numerator. This process consolidates the fractions into a single rational expression.

Our expression, after rewriting with the LCD, is:

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

Now, let's combine the numerators:

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

First, distribute the multiplication:

${\frac{4x + 2 - 2(6x^2 + 3x - 2x - 1) + 5x - 5}{(3x - 1)(x - 1)(2x + 1)}}$

${\frac{4x + 2 - 12x^2 - 2x + 2 + 5x - 5}{(3x - 1)(x - 1)(2x + 1)}}$

Now, combine like terms:

${\frac{-12x^2 + 7x - 1}{(3x - 1)(x - 1)(2x + 1)}}$

Combining the numerators is a crucial step in simplifying rational expressions. It brings together the individual fractions into a single expression, allowing us to further simplify by factoring and canceling common factors. This process highlights the importance of careful distribution and combining like terms to ensure the accuracy of the resulting expression. By consolidating the numerators, we move closer to the final simplified form of the rational expression.

Step 5 Factoring the Numerator and Simplifying

After combining the numerators, the next step is to factor the resulting polynomial, if possible. Factoring the numerator allows us to identify common factors with the denominator, which can then be canceled to further simplify the expression. This step often reveals hidden simplifications and brings us closer to the most concise form of the rational expression.

In our case, the combined numerator is ${-12x^2 + 7x - 1}$. To factor this quadratic trinomial, we can first factor out a -1 to make the leading coefficient positive: ${-(12x^2 - 7x + 1)}$. Now, we look for two numbers that multiply to ${12 \times 1 = 12}$ and add up to -7. These numbers are -4 and -3. We can then rewrite the middle term and factor by grouping:

${-(12x^2 - 4x - 3x + 1)}$ = ${-(4x(3x - 1) - 1(3x - 1))}$ = ${-((4x - 1)(3x - 1))}$

So, the factored numerator is ${-(4x - 1)(3x - 1)}$.

Now, let's rewrite the entire expression with the factored numerator and denominator:

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

We can see a common factor of ${(3x - 1)}$ in both the numerator and the denominator. Canceling this common factor, we get:

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

Factoring the numerator and simplifying is a critical step in reducing the rational expression to its simplest form. It allows us to identify and eliminate common factors, leading to a more concise and manageable expression. This step highlights the importance of mastering factoring techniques and recognizing opportunities for simplification. By carefully factoring and canceling common factors, we arrive at the final simplified form of the rational expression.

Step 6 State Restrictions

As with any simplification of rational expressions, it's crucial to identify and state the restrictions on the variable. Restrictions are values of the variable that would make the original expression undefined, typically due to division by zero. Identifying these restrictions ensures that our simplified expression is equivalent to the original expression for all valid values of the variable.

To determine the restrictions, we must examine the denominators of the original expression and any intermediate expressions formed during the simplification process before any cancellations are made. This is because canceling a factor that could be zero effectively masks the restriction.

In our original expression, ${\frac{2}{3x^2 - 4x + 1} - \frac{2}{x-1} + \frac{5}{6x^2 + x - 1}}$, we factored the denominators as:

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

Setting each of these factors to zero and solving for ${x}$ will give us the restrictions:

1. ${3x - 1 = 0}$ implies ${x = \frac{1}{3}}$.
2. ${x - 1 = 0}$ implies ${x = 1}$.
3. ${2x + 1 = 0}$ implies ${x = -\frac{1}{2}}$.

Therefore, the original expression is undefined when ${x = \frac{1}{3}}$, ${x = 1}$, or ${x = -\frac{1}{2}}$. These restrictions must be stated alongside the simplified expression to ensure its complete and accurate representation.

Stating restrictions is not just a formality; it's an essential part of mathematical rigor. It ensures that the simplified expression is only used for values where the original expression is defined, preserving the integrity of the mathematical statement. By explicitly stating these restrictions, we provide a complete and accurate solution to the simplification problem.

Solution for Problem 2.2

Thus, the simplified expression is:

${\frac{2}{3x^2 - 4x + 1} - \frac{2}{x-1} + \frac{5}{6x^2 + x - 1} = \frac{-(4x - 1)}{(x - 1)(2x + 1)}}$

with the restrictions:

${x \ne \frac{1}{3}, x \ne 1, x \ne -\frac{1}{2}}$

Conclusion

Simplifying algebraic expressions, especially rational expressions, requires a blend of algebraic techniques, attention to detail, and a systematic approach. By mastering the steps of factoring, finding the LCD, combining fractions, canceling common factors, and stating restrictions, you can confidently tackle complex expressions and arrive at their simplest forms. This guide has provided a detailed walkthrough of two such problems, illustrating the key principles and techniques involved. With practice and a solid understanding of these concepts, you can unlock the power of algebraic simplification and excel in your mathematical pursuits.