key term - Greatest Common Factor

Definition

The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is a fundamental concept in elementary algebra that is applicable in various contexts, including whole numbers, fractions, and factoring.

5 Must Know Facts For Your Next Test

1. The GCF is used to simplify fractions by dividing both the numerator and denominator by the GCF.
2. The GCF can be found using the prime factorization method, where the common prime factors are identified and multiplied together.
3. Knowing the GCF is crucial for adding and subtracting fractions with different denominators, as the fractions must be converted to equivalent fractions with a common denominator.
4. The GCF is also an essential tool in factoring by grouping, where it is used to identify the common factors among a set of terms.
5. In the context of special products, the GCF is used to factor expressions such as the difference of two squares and the sum or difference of two cubes.

Review Questions

• Explain how the greatest common factor is used to simplify fractions.
  • The greatest common factor (GCF) is used to simplify fractions by dividing both the numerator and denominator by the GCF. This reduces the fraction to its simplest form by removing any common factors between the numerator and denominator. For example, if the fraction is $\frac{12}{18}$, the GCF of 12 and 18 is 6. Dividing both the numerator and denominator by 6 results in the simplified fraction $\frac{2}{3}$. Simplifying fractions using the GCF is an important skill in working with fractions and is often required in various algebraic contexts.
• Describe how the greatest common factor is used in the process of adding and subtracting fractions with different denominators.
  • When adding or subtracting fractions with different denominators, the fractions must first be converted to equivalent fractions with a common denominator. The least common multiple (LCM) of the denominators is often used to find the common denominator. However, the greatest common factor (GCF) can also be used to simplify the fractions before finding the common denominator. By dividing both the numerator and denominator of each fraction by their GCF, the fractions can be reduced to their simplest form, making the process of finding the common denominator and performing the addition or subtraction more efficient. This use of the GCF is particularly helpful when working with complex fractions or when the denominators have large values.
• Analyze the role of the greatest common factor in the process of factoring by grouping.
  • Factoring by grouping is a technique used to identify and extract common factors from a set of terms. The greatest common factor (GCF) plays a crucial role in this process. By first identifying the GCF of the given terms, the terms can be grouped together based on this common factor. This allows for the factorization of the expression, where the GCF is extracted as a common factor, and the remaining factors are then factored further. The ability to recognize and utilize the GCF is essential in the successful application of factoring by grouping, as it simplifies the factorization process and ensures that the expression is fully factored into its prime factors.

Related terms

Factors:

Factors are the integers that divide a given number without a remainder.

Prime Factorization:

The process of expressing a number as a product of its prime factors.

Least Common Multiple (LCM): The smallest positive integer that is divisible by each of the given integers.