Problem 133 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Step-by-Step Solution

Answer

The statement makes sense. As per definition, the greatest common factor is the largest number that divides exactly into two or more numbers. Factoring it out simplifies the problem. The remaining factor, if not prime, can be further factored, following the same logic.

Step 1: Understand the Statement

In this statement, it is mentioned that first factoring out the greatest common factor makes it easier to determine how to further factor the remaining factor, assuming it is not a prime number. Here, it is important to understand what the greatest common factor is - it is the highest number that divides exactly into two or more numbers. After taking out the greatest common factor, the remaining factor would be smaller and thus potentially simpler to factor further.

Step 2: Analyze the Logic

Let's look at the logic behind this statement. When factoring a number, it is indeed generally easier to first factor out the greatest common factor. This reduces the size of the number you are working with making the problem simpler. Then, if the remaining factor is a composite number (not prime), it can be broken down further. If the remaining factor were prime, it could not be factored further, as prime numbers only have factors of 1 and themselves.

Step 3: Final Evaluation

From the explanation in step 2, we can conclude that the statement makes sense. The process described is a common and effective strategy for factoring numbers.

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a valuable tool in mathematics. It helps simplify expressions and solve problems more efficiently. To find the GCF of two or more numbers, identify the largest number that divides each of them without leaving a remainder. This helps reduce the size of the numbers involved in further calculations.

• For example: Considering the numbers 30 and 45. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 45 are 1, 3, 5, 9, 15, and 45.
• The common factors are 1, 3, 5, and 15. The greatest of these is 15, so the GCF is 15.

After taking out the GCF, the remaining numbers are generally smaller. This simplification makes it easier to work with them in further factoring steps.

Composite Numbers

Composite numbers play a key role in factoring as they can be broken down into smaller factors. Unlike prime numbers, composite numbers have more than two distinct factors. This means they can be divided further beyond just 1 and themselves.

• For example: The number 12 is a composite number because it has factors such as 1, 2, 3, 4, 6, and 12.
• It can be factored into 2 × 6, or 3 × 4, providing multiple ways to break it down.

When factoring a larger number, identifying any composite numbers as factors provides clues for further breaking down the numbers. The more composite numbers you can spot, the more the factoring process is streamlined.

Prime Numbers

Prime numbers are the building blocks of all other numbers. A prime number is defined as having exactly two distinct positive divisors: 1 and itself. This unique property means once a prime number is reached in factoring, no further breakdown is possible.

• For example: The number 5 is a prime number because it can only be divided by 1 and 5 without leaving a remainder.
• Other examples include 2, 3, 7, and 11.

While factoring, encountering a prime number indicates the end of the decomposition process for that path. Recognizing primes quickly in your calculations aids in avoiding unnecessary steps and enhances the efficiency of problem-solving.