Problem 1 - Introductory Algebra for College Students

Question

Find the greatest common factor of each list of monomials. 4 and \(8 x\)

Step-by-Step Solution

Answer

The Greatest Common Factor of 4 and \(8x\) is 4.

Step 1: Identify the coefficients and find their GCF

The monomials given in this exercise are 4 and \(8x\). The coefficients here are 4 and 8. The Greatest Common Factor (GCF) of 4 and 8 is 4.

Step 2: Identify the variables and find their GCF

The first monomial has no variable and the second monomial has \(x\). So the GCF of variables is simply the variable with the lowest power, in this case, there's no variable in first monomial, so it's sufficient to leave it as is.

Step 3: Combine the GCF of coefficients and variables

The GCF of the two monomials, which is the product of the GCF of their coefficients and variables, is 4.

Key Concepts

GCF of Monomials

Finding the greatest common factor (GCF) among monomials is an essential skill in algebra, particularly when simplifying expressions or solving equations. A monomial, put simply, is a single term like 5, \(7x\), or \(3x^2y\) that is composed of numbers, variables, and their powers. To determine the GCF of two or more monomials, you need to break down each one into its prime factors.

For example, consider the exercise where we're asked to find the GCF of 4 and \(8x\). Firstly, we factor each coefficient: 4 is already a prime number, but 8 can be broken down into \(2 \times 2 \times 2\). We only look at coefficients first, ignoring the variables. The common factors of 4 and 8 are 2s, and the GCF of these coefficients is the product of the lowest number of 2s common to both, which is 4.

It's vital not to overlook the simple cases where a monomial does not contain a variable or power – it still represents an important part of the equation. In the absence of common variables, the GCF of the coefficients stands as the GCF of the entire set of monomials.

Coefficients in Algebra

In algebra, a coefficient is the number in front of a variable that indicates how many units of that variable are being considered. Coefficients are foundational in understanding algebraic expressions because they determine the multiplicative factor of the variables they accompany.

In the context of our example, the coefficients of the monomials 4 and \(8x\) are 4 and 8, respectively. The GCF of these coefficients is 4 as previously determined. In this step, the variables were not considered yet; we exclusively focused on the numerical parts. It's crucial to be comfortable with identifying and manipulating coefficients, as they are often the first point of simplification in algebraic problems.

Having a clear understanding of coefficients also helps in further algebraic endeavors like factoring, distributing, or simplifying complex expressions. Without acknowledging the role coefficients play, one might struggle to grasp more advanced algebraic concepts.

Variables in Algebra

Variables are symbols, usually letters, that represent unknown values in algebraic expressions. In our example, \(x\) is the only variable present. When working with monomials, it's important to understand that each variable can have a power, which shows how many times it is multiplied by itself.

However, when a monomial does not have an explicit variable, like the number 4, we can think of it as a variable raised to the zero power. Any number or variable to the zero power is 1, making the concept of a missing variable easier to handle. In our problem, since there is no shared variable between the two monomials – one having \(x\) and the other none – we can consider the lowest power of \(x\) to be 0, effectively treating the first term as \(4x^0\), which is simply 4.

In conclusion, when finding the GCF between variables of monomials, we consider the lowest exponent of common variables. If no common variables exist, the variables are omitted from the GCF. This understanding is critical for simplifying expressions and solving more complex algebraic problems.