Unveiling The Solution To (x-3)(x^2+3x+9) A Step-by-Step Guide

As we delve into the realm of algebra, mastering polynomial multiplication becomes paramount. This article serves as a comprehensive guide to unraveling the intricacies of multiplying the expression $(x-3)(x^2+3x+9)$, providing a step-by-step solution and elucidating the underlying mathematical principles. This is a crucial concept in mathematics, and understanding it thoroughly can significantly enhance your problem-solving skills. This article aims to not only provide the answer but also to explain the process in a clear and understandable manner. We will break down each step, ensuring that you grasp the underlying logic and can apply it to similar problems in the future. Whether you are a student grappling with algebra or simply looking to refresh your mathematical knowledge, this guide will equip you with the tools and understanding you need to confidently tackle polynomial multiplication problems. Let's embark on this mathematical journey together and unlock the solution to $(x-3)(x^2+3x+9).

Deciphering the Problem The Core of Polynomial Multiplication

The given expression, $(x-3)(x^2+3x+9)$, presents a classic example of polynomial multiplication. Our mission is to determine the equivalent simplified form from the options provided: a) $x^3+8$, b) $x^3-27$, c) $x^3+9x-27$, and d) $x^3-4x+27$. To embark on this mathematical journey, we'll employ the distributive property, a fundamental principle that governs polynomial multiplication. This property dictates that each term within the first polynomial must be meticulously multiplied by every term within the second polynomial. This methodical approach ensures that we account for all possible combinations and arrive at the correct result. The distributive property is not merely a mathematical rule; it's a powerful tool that allows us to break down complex expressions into manageable components. By applying this property systematically, we can unravel the intricacies of polynomial multiplication and confidently arrive at the simplified form of the expression. Understanding the distributive property is not just about solving this particular problem; it's about building a solid foundation for more advanced algebraic concepts. This principle is the cornerstone of polynomial manipulation and will serve you well in your mathematical endeavors. Let's now delve into the practical application of this property to the expression at hand.

Step-by-Step Solution A Journey Through Multiplication

Let's embark on the step-by-step solution to multiplying $(x-3)(x^2+3x+9)$.

1. Distribute 'x': We begin by distributing the 'x' from the first term $(x-3)$ across the second polynomial $(x^2+3x+9)$: $x * (x^2+3x+9) = x^3 + 3x^2 + 9x$ This first step is crucial as it lays the groundwork for the rest of the calculation. By multiplying 'x' with each term in the second polynomial, we ensure that 'x' is properly accounted for in the final result. This methodical distribution is the key to accurate polynomial multiplication. Each term resulting from this multiplication contributes to the overall expression, and missing even one term can lead to an incorrect answer. Therefore, meticulousness in this step is paramount.

2. Distribute '-3': Next, we distribute the '-3' from the first term $(x-3)$ across the second polynomial $(x^2+3x+9)$: $-3 * (x^2+3x+9) = -3x^2 - 9x - 27$ Distributing '-3' requires careful attention to signs. The negative sign in front of the '3' will change the sign of each term in the second polynomial upon multiplication. This is a common area for errors, so it's crucial to double-check the signs. The correct distribution of the negative sign is as important as the numerical multiplication. A mistake in the sign will lead to a completely different result. Therefore, pay close attention to the signs as you perform this distribution.

3. Combine Like Terms: Now, we combine the results from steps 1 and 2: $(x^3 + 3x^2 + 9x) + (-3x^2 - 9x - 27)$ Combining like terms is the final simplification step. It involves identifying terms with the same variable and exponent and adding their coefficients. This step is crucial for expressing the polynomial in its most concise form. For instance, $3x^2$ and $-3x^2$ are like terms, as are $9x$ and $-9x$. Combining these terms will simplify the expression significantly.

4. Simplify: By combining like terms, we get: $x^3 + (3x^2 - 3x^2) + (9x - 9x) - 27 = x^3 - 27$ In this step, we observe that the $3x^2$ and $-3x^2$ terms cancel each other out, as do the $9x$ and $-9x$ terms. This leaves us with a much simpler expression: $x^3 - 27$. This simplification is a direct result of the distributive property and the subsequent combination of like terms. The final simplified form represents the equivalent expression to the original product. This simplified form is not only more concise but also easier to work with in further mathematical operations.

Therefore, the result of multiplying $(x-3)(x^2+3x+9)$ is $x^3-27$.

Identifying the Correct Option The Final Verdict

Based on our step-by-step solution, the correct option that corresponds to the result of multiplying $(x-3)(x^2+3x+9)$ is b) $x^3-27$. We arrived at this solution by meticulously applying the distributive property, combining like terms, and simplifying the resulting expression. This process highlights the importance of a systematic approach to solving mathematical problems. Each step, from the initial distribution to the final simplification, plays a crucial role in ensuring the accuracy of the result.

Options a), c), and d) are incorrect because they do not result from the correct application of the distributive property and simplification. These options may represent common mistakes that arise from errors in sign manipulation or the incorrect combination of like terms. Understanding why these options are incorrect is as important as understanding why the correct option is correct. By analyzing the potential errors that lead to these incorrect answers, you can strengthen your understanding of polynomial multiplication and avoid similar mistakes in the future.

The Difference of Cubes A Pattern Revealed

Interestingly, the expression $(x-3)(x^2+3x+9)$ is a specific instance of a broader algebraic identity known as the difference of cubes. The difference of cubes factorization is a powerful tool that allows us to quickly factorize expressions of the form $a^3 - b^3$. The general formula for the difference of cubes is:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

In our case, we can see that $a = x$ and $b = 3$. If we substitute these values into the difference of cubes formula, we get:

$x^3 - 3^3 = (x - 3)(x^2 + 3x + 9)$

$x^3 - 27 = (x - 3)(x^2 + 3x + 9)$

This confirms our earlier result and provides a deeper understanding of the structure of the expression. Recognizing patterns like the difference of cubes can significantly simplify algebraic manipulations. It allows us to bypass the step-by-step multiplication process and arrive at the solution more efficiently. Furthermore, understanding these patterns enhances our ability to factorize and simplify more complex expressions.

Mastering Polynomial Multiplication The Key to Algebraic Success

In conclusion, the correct answer to the question "¿Cuál es la opción que corresponde con el resultado de multiplicar $(x-3)(x^2+3x+9)$?" is b) $x^3-27$. This result was obtained through a systematic application of the distributive property, careful combination of like terms, and a final simplification. Additionally, we explored the connection to the difference of cubes factorization, highlighting the importance of recognizing algebraic patterns.

Mastering polynomial multiplication is a fundamental skill in algebra. It forms the basis for more advanced topics such as factoring, solving equations, and working with rational expressions. By understanding the underlying principles and practicing diligently, you can build a strong foundation in algebra and confidently tackle a wide range of mathematical problems. Remember, mathematics is not just about memorizing formulas; it's about understanding the concepts and applying them creatively. This problem serves as an excellent example of how a systematic approach, combined with a knowledge of algebraic identities, can lead to a successful solution. Continue to explore, practice, and challenge yourself, and you will undoubtedly achieve algebraic success.