Simplifying Algebraic Expressions Step-by-Step Guide

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. This article delves into the process of simplifying the expression $3x(x-2y-5z) + x(x+6y)$. We will break down the steps, explain the underlying principles, and provide a clear, comprehensive guide to help you master this essential mathematical technique.

Understanding the Basics of Algebraic Expressions

Before we dive into the simplification process, it's crucial to understand the basic components of algebraic expressions. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. The expression $3x(x-2y-5z) + x(x+6y)$ contains the variable x, y, and z, along with constants such as 3 and 6. Understanding these basic components is a foundational step in simplifying any algebraic expression.

Step-by-Step Simplification of $3x(x-2y-5z) + x(x+6y)$

Now, let's embark on the simplification journey, step by step. Our expression is $3x(x-2y-5z) + x(x+6y)$. The primary goal here is to eliminate parentheses and combine like terms. This involves applying the distributive property and then identifying and combining terms that share the same variable factors.

1. Applying the Distributive Property

The distributive property is a cornerstone of algebraic simplification. It states that a(b + c) = ab + ac. In other words, to multiply a single term by an expression inside parentheses, you multiply the term by each term within the parentheses individually. Let's apply this to our expression:

  • First, distribute 3x across (x - 2y - 5z): 3x * x = 3x2, 3x * -2y = -6xy, and 3x * -5z = -15xz. This gives us 3x2 - 6xy - 15xz.
  • Next, distribute x across (x + 6y): x * x = x2 and x * 6y = 6xy. This results in x2 + 6xy.

After the first distribution, our expression transforms to 3x2 - 6xy - 15xz + x2 + 6xy. The distributive property allows us to remove the parentheses, paving the way for the next step: combining like terms.

2. Identifying and Combining Like Terms

Like terms are terms that have the same variables raised to the same powers. For instance, 3x2 and x2 are like terms, as are -6xy and 6xy. To simplify further, we combine these like terms by adding or subtracting their coefficients (the numerical part of the term). Let's identify and combine the like terms in our expression:

  • Combine the x2 terms: We have 3x2 and x2. Adding their coefficients (3 + 1) gives us 4x2.
  • Combine the xy terms: We have -6xy and 6xy. Adding their coefficients (-6 + 6) results in 0xy, which is simply 0. These terms effectively cancel each other out.
  • The xz term: The term -15xz has no like terms in the expression, so it remains as is.

After combining like terms, our expression simplifies to 4x2 - 15xz. This is the most simplified form of the original expression.

The Simplified Expression: 4x2 - 15xz

After meticulously applying the distributive property and combining like terms, we arrive at the simplified expression: 4x2 - 15*xz. This concise form is mathematically equivalent to the original, more complex expression. It's easier to work with and understand, highlighting the power of algebraic simplification.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Incorrectly applying the distributive property: Ensure you multiply the term outside the parentheses by every term inside. A missed multiplication can throw off the entire simplification.
  • Combining unlike terms: Remember, you can only combine terms that have the same variables raised to the same powers. Mixing up terms like x2 and x or xy and xz will lead to errors.
  • Sign errors: Pay close attention to the signs (+ and -) when distributing and combining terms. A simple sign mistake can change the entire outcome.
  • Forgetting to distribute negative signs: When distributing a negative term, remember to change the sign of every term inside the parentheses.

Practice Problems

To solidify your understanding, let's work through a couple of practice problems:

  1. Simplify: 2a(a + 3b) - a(2a - b)
  2. Simplify: 5x(x - y) + 2y(x + 3y)

By working through these problems, you'll reinforce your skills and gain confidence in simplifying algebraic expressions.

Conclusion

Simplifying the algebraic expression $3x(x-2y-5z) + x(x+6y)$ involves applying the distributive property and combining like terms. The simplified form, 4x2 - 15*xz, is a testament to the efficiency of algebraic manipulation. By understanding the fundamental principles and practicing regularly, you can master the art of simplifying algebraic expressions. This skill is not only crucial for success in mathematics but also for various fields that rely on mathematical modeling and problem-solving.

Remember, algebraic simplification is not just about finding the right answer; it's about developing a deeper understanding of mathematical relationships. So, keep practicing, stay curious, and embrace the beauty of mathematics!