Simplifying Algebraic Expressions A Step-by-Step Guide For 3x(x-2y-5z) + X(x+6y)
In the realm of mathematics, algebraic expressions form the bedrock of problem-solving and analytical reasoning. This article delves into the intricacies of simplifying a given algebraic expression, 3x(x-2y-5z) + x(x+6y). Through a step-by-step approach, we will unravel its components, apply the principles of distribution and combining like terms, and arrive at the most simplified form. Understanding the underlying concepts and techniques is paramount in mastering algebra and paving the way for more advanced mathematical explorations.
Dissecting the Expression: Initial Assessment
The expression at hand, 3x(x-2y-5z) + x(x+6y), is a combination of terms involving variables x, y, and z. To effectively simplify it, we need to identify the operations involved and the order in which they must be performed. The expression primarily involves multiplication and addition, with parentheses indicating the order of operations. The distributive property will be crucial in expanding the terms and removing the parentheses. Before diving into the simplification process, let's outline the key steps involved:
- Distribution: Apply the distributive property to multiply the terms outside the parentheses with the terms inside.
- Expansion: Expand the expression by writing out all the resulting terms.
- Combining Like Terms: Identify and combine terms with the same variables and exponents.
- Simplification: Present the expression in its most concise and simplified form.
By following these steps systematically, we can transform the complex expression into a more manageable and understandable form. The ability to simplify algebraic expressions is not only fundamental in mathematics but also essential in various fields such as physics, engineering, and computer science, where mathematical models are used to represent and solve real-world problems.
Applying the Distributive Property
The distributive property is a cornerstone of algebraic manipulation. It states that for any numbers a, b, and c, a(b + c) = ab + ac. This property allows us to multiply a single term by a group of terms enclosed in parentheses. In our expression, 3x(x-2y-5z) + x(x+6y), we have two instances where we need to apply the distributive property. Let's break it down step by step.
First, consider the term 3x(x-2y-5z). We need to distribute the 3x to each term inside the parentheses:
- 3x * x = 3x²
- 3x * (-2y) = -6xy
- 3x * (-5z) = -15xz
Combining these results, we get 3x² - 6xy - 15xz.
Next, let's apply the distributive property to the term x(x+6y):
- x * x = x²
- x * (6y) = 6xy
This gives us x² + 6xy. Now, we can rewrite the original expression with the parentheses removed:
3x(x-2y-5z) + x(x+6y) = 3x² - 6xy - 15xz + x² + 6xy
The application of the distributive property has expanded the expression, making it easier to identify and combine like terms. This step is crucial in simplifying complex algebraic expressions and laying the groundwork for further manipulation. The distributive property is not just a mathematical rule; it is a tool that allows us to break down complex problems into smaller, more manageable parts.
Combining Like Terms: The Art of Grouping
After applying the distributive property, our expression stands as 3x² - 6xy - 15xz + x² + 6xy. The next step in simplifying this expression is to combine like terms. Like terms are those that have the same variables raised to the same powers. In our expression, we can identify the following pairs of like terms:
- 3x² and x²: These terms both have the variable x raised to the power of 2.
- -6xy and 6xy: These terms both have the variables x and y, each raised to the power of 1.
To combine like terms, we simply add or subtract their coefficients (the numerical part of the term). Let's start with the x² terms:
3x² + x² = (3 + 1)x² = 4x²
Next, let's combine the xy terms:
-6xy + 6xy = (-6 + 6)xy = 0xy = 0
The xy terms cancel each other out, leaving us with no xy term in the simplified expression. The remaining term, -15xz, does not have any like terms, so it remains as it is. Now, we can rewrite the expression by combining the like terms:
3x² - 6xy - 15xz + x² + 6xy = 4x² - 15xz
Combining like terms is a fundamental skill in algebra. It allows us to consolidate multiple terms into a single term, making the expression more concise and easier to work with. The process involves identifying terms with the same variable composition and then performing the arithmetic operations on their coefficients. This step is essential in simplifying algebraic expressions and solving equations.
The Simplified Expression: A Concise Form
After applying the distributive property and combining like terms, we have arrived at the simplified form of the expression: 4x² - 15xz. This expression is much more concise and easier to understand than the original expression, 3x(x-2y-5z) + x(x+6y). The simplification process has effectively reduced the number of terms and eliminated redundancies, resulting in a more elegant and manageable expression.
The simplified expression, 4x² - 15xz, contains only two terms, each with a distinct variable composition. The first term, 4x², represents a quadratic term involving the variable x, while the second term, -15xz, represents a term involving the product of variables x and z. This simplified form allows us to easily identify the components of the expression and their relationships.
The process of simplifying algebraic expressions is not merely an exercise in mathematical manipulation; it is a crucial skill that underpins many areas of mathematics and its applications. A simplified expression is easier to analyze, evaluate, and manipulate further. It also provides a clearer understanding of the underlying relationships between the variables involved. In this case, the simplified expression, 4x² - 15xz, provides a concise representation of the original expression and facilitates further mathematical operations, such as solving equations or graphing functions.
Conclusion: Mastering Algebraic Simplification
In this article, we have embarked on a journey to simplify the algebraic expression 3x(x-2y-5z) + x(x+6y). Through a systematic application of the distributive property and the process of combining like terms, we have successfully transformed the expression into its simplified form: 4x² - 15xz. This process highlights the fundamental principles of algebraic manipulation and their importance in simplifying complex mathematical expressions.
The ability to simplify algebraic expressions is a cornerstone of mathematical proficiency. It not only enhances our understanding of algebraic structures but also provides a foundation for more advanced mathematical concepts. The techniques we have employed in this article, such as the distributive property and combining like terms, are applicable to a wide range of algebraic expressions and problems.
By mastering these techniques, we can confidently tackle complex mathematical challenges and unlock the power of algebraic reasoning. The journey of simplifying algebraic expressions is not just about finding the right answer; it is about developing a deeper understanding of mathematical principles and honing our problem-solving skills. The simplified expression, 4x² - 15xz, stands as a testament to the power of algebraic simplification and its role in unveiling the elegance and conciseness of mathematical expressions. Embracing these skills will undoubtedly pave the way for continued success in mathematics and its myriad applications.
