Understanding how to factor algebraic expressions is essential for students and professionals alike. In this article, we will explore the expression 6x²(x - 2) + 7(x - 2) and demonstrate how to arrive at its completely factored form. This topic is not only relevant for academic purposes but is also crucial for applications in various fields such as engineering, physics, and economics. By breaking down this expression, we aim to provide you with a clear and comprehensive understanding of the factoring process.
Factoring is a fundamental concept in algebra that allows us to simplify expressions and solve equations more efficiently. The expression we are examining today combines both polynomial terms and a common factor, making it an excellent example to illustrate the factoring process. We will go through each step methodically, ensuring that you leave with a solid grasp of how to tackle similar problems in the future.
In this article, we will cover the following topics: the importance of factoring, step-by-step factoring of the expression, and some practical applications of the factored form. Through careful explanation and examples, we hope to enhance your understanding of algebraic expressions and their factored forms. Let’s dive in!
Table of Contents
1. Importance of Factoring
Factoring is a crucial skill in algebra that simplifies expressions and solves equations efficiently. Here are some reasons why factoring is important:
- It helps in simplifying complex algebraic expressions.
- It is essential for solving quadratic equations.
- Factored forms provide insights into the roots and behaviors of functions.
- It is widely used in higher-level mathematics and various scientific fields.
2. Step-by-Step Factoring of 6x²(x - 2) + 7(x - 2)
2.1 Identifying Common Factors
To factor the expression 6x²(x - 2) + 7(x - 2), the first step is to identify the common factors in both terms. Notice that both terms contain the factor (x - 2). This allows us to factor (x - 2) out of the expression:
2.2 Factoring Out the Common Term
We can write the expression as follows:
6x²(x - 2) + 7(x - 2) = (x - 2)(6x² + 7)
Now, we have factored out the common term (x - 2). The remaining expression, 6x² + 7, does not have any further common factors or can be factored using real numbers, which means this is the completely factored form.
3. Resulting Factored Form
The completely factored form of the expression 6x²(x - 2) + 7(x - 2) is:
(x - 2)(6x² + 7)
This form is useful because it reveals the roots of the equation when set to zero, providing insight into the behavior of the polynomial function.
4. Practical Applications of Factored Forms
Understanding the factored form of polynomial expressions is not just an academic exercise. Here are some real-world applications:
- Engineering: Factoring is used in structural analysis to determine load distributions.
- Physics: In kinematics, factored equations help solve motion problems.
- Economics: Factored forms can simplify the analysis of cost functions and profit maximization.
5. Conclusion
In this article, we have explored the expression 6x²(x - 2) + 7(x - 2) and demonstrated how to arrive at its completely factored form, (x - 2)(6x² + 7). We discussed the significance of factoring in algebra and its various applications in real-world scenarios. Understanding how to factor expressions is a vital skill that enhances problem-solving abilities in mathematics and beyond.
We encourage you to practice factoring more expressions and to share your thoughts in the comments below. If you found this article helpful, consider sharing it with your peers or exploring other related articles on our site for more insights into algebra and mathematics.
Thank you for reading, and we hope to see you again soon!
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