Problem 16 - Elementary Algebra

Question

Find the greatest common factor. \(20 y^{3}, 28 y^{2}, 40 y\)

Step-by-Step Solution

Answer

The GCF is 4y.

Step 1 - Find the prime factorizations

Break down each coefficient into its prime factors. The coefficients in your terms are 20, 28, and 40. Their factorizations are: 20: 2² × 5 28: 2² × 7 40: 2³ × 5

Step 2 - Identify the common factors

List the factors that appear in all the factorizations. Here, each term has at least 2² in common.

Step 3 - Find the lowest power of the common factors

For the 2's, the lowest power is 2². There are no other prime factors common to all coefficients.

Step 4 - Determine the common factor for variables

Each term has the variable y, with the lowest exponent being y in the term 40y.

Step 5 - Compute the greatest common factor (GCF)

Combine the lowest powers of the common factors and variables: 2² = 4 and y. Thus, the GCF is 4y.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the number 20 can be broken down prime factors as follows:

• 20: 2² × 5
• 28: 2² × 7
• 40: 2³ × 5

In each case, we are finding the prime numbers that, when multiplied together, give us the original number. This process helps in determining the greatest common factor (GCF) because it allows us to easily identify and compare the components of different numbers. The greatest common factor will include the lowest powers of all prime factors common to each term.

Common Factors

A common factor is a factor that two or more numbers share. When finding the greatest common factor, it's important to identify which factors the numbers have in common. In our example, after breaking down the coefficients of each term, we found the following:

• 20: 2² × 5
• 28: 2² × 7
• 40: 2³ × 5

The common prime factors here are the 2's. Specifically, each number has at least two 2's in its factorization, or 2². By identifying and considering only the common factors, we simplify the process of finding the GCF, which is the largest factor shared by all original numbers.

Variables

In algebra, variables are symbols that represent numbers. When finding the greatest common factor of algebraic terms, we must also consider the variables. In our exercise, each term contains the variable \(y\):

• 20y³
• 28y²
• 40y

We look at the lowest exponent of the variable common to all terms. Here, the lowest exponent is 1, from the term 40y. Therefore, the variable part of the GCF will be y.
Combining this with the common prime factors we identified (which was 4 from 2²), the GCF in this example is 4y. Recognizing how variables factor into the GCF is essential for solving more complex algebraic expressions.