Problem 16 - Elementary Algebra

Question

Determine the GCF of all the terms. $$ 12 x 3,4 x 2,6 x $$

Step-by-Step Solution

Answer

The GCF of the terms is \(2x\).

Step 1: Understand the Problem

We need to find the greatest common factor (GCF) of the numbers 12, 4, and 6, and the common factor including the variable in the terms that contain "\(x\)".

Step 2: Break Down Numerical Part

Identify the numerical part of each term: 12, 4, and 6. We'll find the GCF of these numbers.

Step 3: Find Prime Factorization

Prime factorize each number: 12 is \(2^2 \times 3\), 4 is \(2^2\), and 6 is \(2 \times 3\).

Step 4: Identify Common Factors

Identify the smallest power of each common prime factor. The common factor is \(2^1\), as all numbers share at least one 2.

Step 5: Determine GCF of Numbers

The GCF for 12, 4, and 6 is 2, since it is the highest number that divides all of them.

Step 6: Consider Variable Terms

Identify the \(x\) terms: \(x^3\), \(x^2\), and \(x\). The lowest power of \(x\) is \(x^1\).

Step 7: Combine Numerical and Variable GCF

The GCF of the full expression includes both the numerical part and the variables. Thus, it is \(2x\).

Key Concepts

Prime Factorization

Prime factorization is a process where we break down a number into its prime numbers. A prime number is a number greater than 1 with no divisors other than 1 and itself. By breaking numbers into prime factors, we can easily spot common factors among different numbers.
In the given problem, we have three numbers: 12, 4, and 6. Let's look into how we break each into its prime components:

• 12 can be expressed as \(2^2 \times 3\)
• 4 can be expressed as \(2^2\)
• 6 can be expressed as \(2 \times 3\)

After factoring, we compare these expressions to identify the smallest power of each common prime factor.

Variables in Algebra

In algebra, variables are symbols, typically letters, used to represent numbers. They allow for general solutions and expressing relationships mathematically. In this specific scenario, we're dealing with the variable \(x\), which appears in different powers.
The terms given in the exercise include:

• \(x^3\) in the first term
• \(x^2\) in the second term
• \(x\) in the third term

When determining the greatest common factor (GCF), we need to find the smallest power of the variable present in all terms. Here, since \(x\) appears as \(x\) in the third term, we determine the common variable factor to be \(x^1\).

Numerical Coefficients

Numerical coefficients are the constant numbers that multiply the variables in algebraic expressions. In the given problem, the coefficients are the numbers that prelude the variable component.
Let's highlight the numerical coefficients in each term:

• For \(12x^3\), the numerical coefficient is 12
• For \(4x^2\), the numerical coefficient is 4
• For \(6x\), the numerical coefficient is 6

Finding the GCF among these numerical parts involves first using prime factorization and then determining the highest common divisor. The GCF identified for the coefficients 12, 4, and 6 is 2.

Elementary Algebra

Elementary algebra involves the basic manipulation of algebraic expressions to solve for unknowns. One fundamental aspect is determining common factors, like in this exercise. This step is crucial in simplifying expressions and solving equations efficiently.
Here, to find the GCF of the terms \(12x^3, 4x^2,\) and \(6x\), we break each term into its numerical and variable components. After prime factorizing and identifying the smallest power of the variable, we combine the GCF of numerical and variable parts.
The overall GCF in this exercise is determined to be \(2x\), with 2 coming from the coefficients and \(x\) from the variables. Understanding and applying elementary algebra helps bring clarity and structure to solving such problems.