Simplify (3x^3 - 27x + 54) / (3x - 3) A Step-by-Step Guide

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When dealing with rational expressions, simplification is a crucial step in making them easier to work with. This article aims to provide a detailed guide on how to simplify the rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$ into the form ${ q(x) + \frac{r(x)}{b(x)} }$. We will explore the methods and steps involved, ensuring a clear understanding of the process. This article delves into simplifying the rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$, a fundamental task in algebra. Rational expressions are fractions where the numerator and denominator are polynomials. Simplifying such expressions often involves factoring, polynomial long division, and reducing the fraction to its simplest form. The goal is to rewrite the given expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) }$ is the quotient, ${ r(x) }$ is the remainder, and ${ b(x) }$ is the divisor. This form is particularly useful in calculus and other advanced mathematical contexts.

Before diving into the simplification process, it’s important to understand the basic principles of polynomial factorization and division. Factoring involves breaking down a polynomial into its constituent factors, while polynomial long division is a method for dividing one polynomial by another. These techniques are essential for simplifying rational expressions and expressing them in a more manageable form. The process typically involves several key steps, including factoring the numerator and the denominator, identifying common factors, dividing the polynomials, and expressing the result in the desired form. Each step requires a solid understanding of algebraic principles and techniques.

In the subsequent sections, we will walk through each step in detail, providing clear explanations and examples to illustrate the process. Whether you are a student learning algebra or a professional looking to refresh your skills, this guide will provide you with the knowledge and tools needed to simplify rational expressions effectively. By the end of this article, you will have a comprehensive understanding of how to simplify complex rational expressions and express them in their simplest form. This skill is not only crucial for academic success but also for practical applications in various fields of mathematics and engineering. Let’s begin by examining the first step: factoring the numerator and the denominator of the given rational expression.

Step-by-Step Simplification

1. Factoring the Numerator and Denominator

To simplify the rational expression, the first step involves factoring both the numerator and the denominator. Factoring simplifies the expression by breaking down complex polynomials into simpler terms. This process makes it easier to identify common factors that can be cancelled out. This section focuses on the crucial first step in simplifying the rational expression: factoring the numerator and the denominator. Factoring is a fundamental technique in algebra that allows us to break down polynomials into simpler, more manageable terms. This process is essential for simplifying rational expressions, as it helps to identify common factors that can be cancelled out, leading to a simpler form of the expression.

The given numerator is ${ 3x^3 - 27x + 54 }$. We can start by factoring out the common factor of 3:

${ 3(x^3 - 9x + 18) }$

Next, we need to factor the cubic polynomial ${ x^3 - 9x + 18 }$. This can be done by looking for roots of the polynomial. By trial and error, or using the Rational Root Theorem, we can find that ${ x = 3 }$ is a root. Therefore, ${ (x - 3) }$ is a factor. Now, we can perform synthetic division or polynomial long division to divide ${ x^3 - 9x + 18 }$ by ${ (x - 3) }$. Alternatively, we can attempt to factor the cubic expression by inspection or by using factoring techniques suitable for cubic polynomials. The goal is to find factors that, when multiplied, yield the original cubic expression.

After dividing ${ x^3 - 9x + 18 }$ by ${ (x - 3) }$, we get ${ x^2 + 3x - 6 }$. This quadratic expression can be further factored. We look for two numbers that multiply to -6 and add to 3. These numbers are 6 and -3. Thus, we can rewrite the quadratic as ${ (x+6)(x-3) }$. Therefore, the factored form of the numerator is:

${ 3(x - 3)(x + 6) }$

For the denominator, ${ 3x - 3 }$, we can simply factor out the common factor of 3:

${ 3(x - 1) }$

Factoring the numerator and denominator is a critical step in simplifying rational expressions. It allows us to identify common factors that can be cancelled out, which is the next step in the simplification process. The ability to factor polynomials efficiently is a valuable skill in algebra, and it forms the foundation for many advanced mathematical concepts. With the numerator and denominator now factored, we can proceed to the next step: identifying and cancelling common factors. This will bring us closer to expressing the rational expression in its simplest form.

2. Identifying and Cancelling Common Factors

Identifying and cancelling common factors is a pivotal step in simplifying rational expressions. Once the numerator and denominator are factored, we look for factors that appear in both. Cancelling these common factors reduces the expression to its simplest form. In this section, we will focus on how to identify and cancel common factors in the factored rational expression. This step is crucial for simplifying rational expressions, as it eliminates redundant terms and reduces the expression to its most basic form. After factoring both the numerator and the denominator, the next logical step is to look for common factors that can be cancelled out.

From the previous step, we have the factored form of the rational expression:

${ \frac{3(x - 3)(x + 6)}{3(x - 1)} }$

We can see that there is a common factor of 3 in both the numerator and the denominator. By cancelling out this common factor, we simplify the expression to:

${ \frac{(x - 3)(x + 6)}{(x - 1)} }$

Now, we examine the remaining factors to see if there are any other common factors. In this case, there are no other factors that appear in both the numerator and the denominator. The numerator has factors ${ (x - 3) }$ and ${ (x + 6) }$, while the denominator has a factor ${ (x - 1) }$. Since none of these factors are common, we have simplified the expression as much as possible through cancellation. The process of identifying and cancelling common factors is a straightforward yet essential technique in simplifying rational expressions. It relies on the fundamental principle that dividing both the numerator and the denominator by the same non-zero factor does not change the value of the expression. This step is particularly useful when dealing with complex rational expressions, as it reduces the complexity and makes the expression easier to work with.

Once the common factors have been cancelled, the resulting expression is often in a simpler form, but it may still need further simplification depending on the specific requirements of the problem. In this case, we have simplified the rational expression by cancelling the common factor of 3. The next step involves dividing the remaining polynomial in the numerator by the polynomial in the denominator to express the rational expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$. This will provide us with the final simplified form of the given expression. Let’s move on to the next step: performing polynomial long division to further simplify the expression.

3. Performing Polynomial Long Division

Polynomial long division is used when the degree of the numerator is greater than or equal to the degree of the denominator. This process helps to express the rational expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) }$ is the quotient and ${ r(x) }$ is the remainder. This section delves into the process of polynomial long division, a crucial technique for simplifying rational expressions where the degree of the numerator is greater than or equal to the degree of the denominator. Polynomial long division allows us to express the rational expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) }$ is the quotient, ${ r(x) }$ is the remainder, and ${ b(x) }$ is the divisor. This form is particularly useful in various mathematical contexts, including calculus and algebra.

After cancelling common factors, we have the expression:

${ \frac{(x - 3)(x + 6)}{(x - 1)} }$

First, we expand the numerator:

${ (x - 3)(x + 6) = x^2 + 6x - 3x - 18 = x^2 + 3x - 18 }$

Now, we perform polynomial long division to divide ${ x^2 + 3x - 18 }$ by ${ (x - 1) }$. Polynomial long division is similar to the long division of numbers, but instead of digits, we are dealing with terms of polynomials. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the result by the divisor, subtracting from the dividend, and bringing down the next term. This process is repeated until the degree of the remainder is less than the degree of the divisor.

When we divide ${ x^2 + 3x - 18 }$ by ${ (x - 1) }$, we get:

• Quotient: ${ x + 4 }$
• Remainder: ${ -14 }$

Therefore, the result of the polynomial long division can be written as:

${ x + 4 + \frac{-14}{x - 1} }$

Polynomial long division is a fundamental skill in algebra and is essential for simplifying rational expressions. It allows us to break down complex rational expressions into simpler forms, making them easier to analyze and work with. The process requires a systematic approach and a good understanding of polynomial arithmetic. With the quotient and remainder obtained, we can now express the original rational expression in the desired form.

In this case, we have successfully performed polynomial long division and expressed the rational expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$. The quotient ${ q(x) }$ is ${ x + 4 }$, the remainder ${ r(x) }$ is ${ -14 }$, and the divisor ${ b(x) }$ is ${ x - 1 }$. The next step is to write the final simplified form of the rational expression. This will provide us with the ultimate result of our simplification process. Let’s proceed to the final step: expressing the simplified form of the rational expression.

4. Expressing the Simplified Form

After performing polynomial long division, we can now express the rational expression in its simplified form. This form is written as ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) }$ is the quotient, ${ r(x) }$ is the remainder, and ${ b(x) }$ is the divisor. This section focuses on the final step: expressing the simplified form of the rational expression after performing polynomial long division. The goal is to write the expression in the form ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) }$ is the quotient, ${ r(x) }$ is the remainder, and ${ b(x) }$ is the divisor. This form is particularly useful in calculus and other advanced mathematical applications, as it separates the polynomial part from the fractional part of the expression.

From the previous steps, we have:

• Quotient: ${ q(x) = x + 4 }$
• Remainder: ${ r(x) = -14 }$
• Divisor: ${ b(x) = x - 1 }$

Now, we can express the simplified form of the rational expression as:

${ x + 4 + \frac{-14}{x - 1} }$

This is the simplified form of the given rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$. The expression is now in the form ${ q(x) + \frac{r(x)}{b(x)} }$, which is often easier to work with in various mathematical contexts. The process of simplifying rational expressions involves several key steps, including factoring, cancelling common factors, and performing polynomial long division. Each step requires a solid understanding of algebraic principles and techniques. By following these steps, we can effectively simplify complex rational expressions and express them in their simplest form.

Expressing the simplified form is the culmination of the entire simplification process. It provides a clear and concise representation of the original rational expression, making it easier to analyze and manipulate. This final form is not only useful for academic purposes but also for practical applications in various fields of science and engineering. In this case, we have successfully simplified the rational expression and expressed it in the desired form. This completes our step-by-step guide on simplifying rational expressions.

Final Simplified Form

The simplified form of the rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$ is:

${ x + 4 - \frac{14}{x - 1} }$

This comprehensive guide has walked you through each step of the simplification process, from factoring the numerator and denominator to performing polynomial long division and expressing the final simplified form. By mastering these techniques, you can effectively simplify rational expressions and tackle more complex algebraic problems. This section reaffirms the final simplified form of the given rational expression, providing a clear and concise answer to the problem. After completing all the necessary steps, including factoring, cancelling common factors, and performing polynomial long division, we have arrived at the simplified form of the rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$.

The simplified form is:

${ x + 4 - \frac{14}{x - 1} }$

This expression is in the form ${ q(x) + \frac{r(x)}{b(x)} }$, where ${ q(x) = x + 4 }$ is the quotient, ${ r(x) = -14 }$ is the remainder, and ${ b(x) = x - 1 }$ is the divisor. This form is particularly useful in various mathematical contexts, as it separates the polynomial part from the fractional part of the expression. The final simplified form is the result of a series of algebraic manipulations, each of which is crucial for reducing the expression to its simplest terms. Factoring the numerator and denominator allows us to identify common factors that can be cancelled out. Polynomial long division enables us to divide the numerator by the denominator and express the result as a quotient and a remainder.

The ability to simplify rational expressions is a fundamental skill in algebra, and it is essential for success in more advanced mathematical topics. This skill is not only important for academic purposes but also for practical applications in various fields of science and engineering. By mastering the techniques outlined in this guide, you can effectively simplify complex rational expressions and express them in their simplest form. The final simplified form is the ultimate goal of the simplification process. It provides a clear and concise representation of the original rational expression, making it easier to analyze and manipulate. This form is particularly useful when solving equations, graphing functions, and performing other mathematical operations. In this case, we have successfully simplified the rational expression and expressed it in the desired form. This concludes our detailed guide on simplifying the rational expression ${ \frac{3x^3 - 27x + 54}{3x - 3} }$.