Factoring The Greatest Common Factor A Detailed Guide To 63f + 70

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Factoring the greatest common factor (GCF) is a fundamental concept in algebra, serving as a crucial stepping stone for more advanced mathematical manipulations. In this comprehensive guide, we will delve into the process of factoring the expression 63f + 70. We'll break down each step, ensuring clarity and understanding, and provide practical insights to help you master this essential skill.

Understanding the Greatest Common Factor (GCF)

Before we dive into the specifics of factoring 63f + 70, it's crucial to grasp the concept of the greatest common factor (GCF). The GCF, also known as the greatest common divisor (GCD), is the largest number that divides evenly into two or more numbers. In simpler terms, it's the biggest factor that all the terms in an expression share.

To illustrate this, consider the numbers 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors are 1, 2, 3, and 6, with 6 being the greatest among them. Therefore, the GCF of 12 and 18 is 6.

The GCF is not limited to numbers alone; it can also involve variables. For instance, in the expression 4x² + 6x, both terms share a common factor of 2x. Identifying the GCF is the cornerstone of factoring, as it allows us to simplify expressions and solve equations more effectively.

Steps to Find the GCF

Finding the GCF involves a systematic approach:

  1. List the factors: Begin by listing all the factors of each term in the expression. Factors are the numbers or variables that divide evenly into a term.
  2. Identify common factors: Compare the lists of factors and identify the factors that are common to all terms.
  3. Determine the greatest common factor: From the common factors, select the largest one. This is the GCF.

Let's apply these steps to our expression, 63f + 70, to find its GCF.

Factoring 63f + 70: A Step-by-Step Approach

Now that we have a firm understanding of the GCF, let's tackle the task of factoring 63f + 70. We will follow a step-by-step approach to ensure clarity and precision.

Step 1: Identify the Terms

The first step is to identify the individual terms in the expression. In 63f + 70, we have two terms: 63f and 70.

Step 2: Find the GCF of the Coefficients

The coefficients are the numerical parts of the terms. In this case, the coefficients are 63 and 70. To find the GCF of 63 and 70, we'll list their factors:

  • Factors of 63: 1, 3, 7, 9, 21, 63
  • Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

Comparing the lists, we see that the common factors are 1 and 7. The greatest among these is 7. Therefore, the GCF of 63 and 70 is 7.

Step 3: Identify Common Variables

Next, we examine the variables in each term. The term 63f contains the variable 'f', while the term 70 has no variable. Since 'f' is not present in both terms, there is no common variable factor.

Step 4: Determine the Overall GCF

To find the overall GCF of the expression, we combine the GCF of the coefficients and the common variables. In this case, the GCF of the coefficients is 7, and there are no common variables. Therefore, the overall GCF of 63f + 70 is 7.

Step 5: Factor out the GCF

Now that we've identified the GCF, we can factor it out of the expression. This involves dividing each term by the GCF and writing the expression in the form A(Bf + C), where A is the GCF.

Divide each term by 7:

  • 63f / 7 = 9f
  • 70 / 7 = 10

Now, we rewrite the expression by placing the GCF (7) outside the parentheses and the results of the division inside:

7(9f + 10)

This is the factored form of 63f + 70. We have successfully factored out the greatest common factor.

Verifying the Result

To ensure the accuracy of our factoring, we can distribute the GCF back into the parentheses and see if we obtain the original expression. Distributing 7 in 7(9f + 10) gives us:

7 * 9f + 7 * 10 = 63f + 70

This matches our original expression, confirming that our factoring is correct.

Expressing the Answer in the Form A(Bf + C)

The question specifically asks for the answer in the form A(Bf + C), where A, B, and C are numbers. In our factored expression, 7(9f + 10), we can identify:

  • A = 7
  • B = 9
  • C = 10

Thus, the factored form of 63f + 70 in the requested format is 7(9f + 10).

Examples and Practice Problems

To solidify your understanding, let's explore some additional examples and practice problems.

Example 1: Factor 24x + 36

  1. Terms: 24x and 36
  2. GCF of coefficients (24 and 36): 12
  3. Common variables: None
  4. Overall GCF: 12
  5. Factoring: 12(2x + 3)

Example 2: Factor 15a² - 25a

  1. Terms: 15a² and -25a
  2. GCF of coefficients (15 and 25): 5
  3. Common variables: a
  4. Overall GCF: 5a
  5. Factoring: 5a(3a - 5)

Practice Problems:

  1. Factor 32y + 48
  2. Factor 18b² - 27b
  3. Factor 45c + 60

By working through these examples and practice problems, you'll gain confidence in your ability to factor expressions using the greatest common factor.

Importance of Factoring the GCF

Factoring the GCF is not just a mathematical exercise; it's a fundamental skill with numerous applications in algebra and beyond. Here are some key reasons why mastering this technique is essential:

  1. Simplifying Expressions: Factoring the GCF simplifies complex expressions, making them easier to work with. This is particularly useful when solving equations or performing other algebraic manipulations. For example, simplifying 63f + 70 to 7(9f + 10) makes it easier to analyze and manipulate the expression in further calculations.

  2. Solving Equations: Factoring is a crucial step in solving many types of equations, especially quadratic equations. By factoring an equation, we can often find its solutions more readily. For instance, if an equation can be factored into the form A(Bf + C) = 0, then either A = 0 or Bf + C = 0, which can lead to straightforward solutions.

  3. Reducing Fractions: Factoring the GCF can help simplify fractions by canceling out common factors in the numerator and denominator. This is a common practice in algebra and calculus to make expressions more manageable. For example, if we have a fraction like (63f + 70) / 14, factoring the numerator gives us 7(9f + 10) / 14, which simplifies to (9f + 10) / 2 after canceling the common factor of 7.

  4. Advanced Mathematics: Factoring the GCF is a foundational skill for more advanced mathematical topics such as calculus, differential equations, and linear algebra. Many techniques in these areas rely on the ability to factor expressions efficiently and accurately.

  5. Real-World Applications: Factoring skills are applicable in various real-world scenarios, including engineering, physics, and computer science. For instance, in engineering, factoring can be used to simplify equations that model physical systems, making them easier to analyze and design.

Common Mistakes to Avoid

While factoring the GCF is a straightforward process, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accuracy.

  1. Missing the GCF: One of the most common errors is failing to identify the correct GCF. This can happen if you don't list all the factors or overlook a common factor. Always double-check your lists and make sure you've found the greatest common factor.

  2. Incorrect Division: When factoring out the GCF, it's essential to divide each term correctly. A simple arithmetic error can lead to an incorrect factored expression. Take your time and verify your divisions.

  3. Forgetting the Remainder: After dividing each term by the GCF, make sure to include the results within the parentheses. For example, when factoring 63f + 70, dividing by 7 gives us 9f + 10, which should be placed inside the parentheses: 7(9f + 10). Forgetting to include these remainders is a common mistake.

  4. Incorrect Sign: Pay close attention to the signs of the terms. If a term is negative, make sure to include the negative sign in the factored expression. For example, if you're factoring 15a² - 25a, the factored form is 5a(3a - 5), not 5a(3a + 5).

  5. Not Distributing to Check: Always distribute the GCF back into the parentheses to verify your factoring. This simple step can catch errors and ensure that your factored expression is equivalent to the original one. If you don't get back the original expression, you've made a mistake.

Conclusion

In this comprehensive guide, we've explored the process of factoring the greatest common factor (GCF), using 63f + 70 as our primary example. We've broken down each step, from identifying the terms and finding the GCF to factoring out the GCF and verifying the result. We've also discussed the importance of factoring the GCF in algebra and beyond, highlighted common mistakes to avoid, and provided additional examples and practice problems.

Mastering the technique of factoring the GCF is a critical skill in algebra. It simplifies expressions, aids in solving equations, and forms the foundation for more advanced mathematical concepts. By following the steps outlined in this guide and practicing regularly, you'll develop the confidence and proficiency needed to excel in factoring and other algebraic tasks.

Remember, mathematics is a skill that improves with practice. So, keep practicing, keep exploring, and keep mastering the fundamentals. Happy factoring!