Problem 10 - Introductory Algebra for College Students

Question

Find the greatest common factor of each list of monomials. $$x^{2} y, 3 x^{3} y, \text { and } 6 x^{2}$$

Step-by-Step Solution

Answer

The greatest common factor of the monomials \(x^{2} y, 3 x^{3} y\), and \(6 x^{2}\) is \(x^{2}\)

Step 1: Identify the Common Factors

The common factors in \(x^{2} y, 3 x^{3}y\) and \(6 x^{2}\) are \(x^{2}\) and \(1\). The exponents on \(x\) in all of the monomials are 2, 3, and 2 respectively. The smallest exponent is 2, so the common factor for \(x\) is \(x^{2}\). None of the monomials have \(y\) as a common factor, so the greatest common factor is \(x^{2}\).

Step 2: Write the GCF

The greatest common factor of the given monomials \(x^{2} y, 3 x^{3}y\) and \(6 x^{2}\) is \(x^{2}\).

Key Concepts

Monomials

Understanding monomials is crucial when exploring algebraic expressions and finding their greatest common factor (GCF). A monomial is a single term in an algebraic expression that consists of a coefficient and a variable. The variable may have an exponent, representing repeated multiplication of that variable. For example, in the expression x^2y, x^2 is a monomial, with x being the variable and 2 being the exponent indicating that x is used twice in multiplication.

When identifying the GCF among monomials, it's important to consider both the numerical coefficients and the variables raised to powers. In our exercise, the list of monomials provided (x^2 y, 3x^3y, and 6x^2) differ by coefficients and the variable y, but share a common base of the variable x with a qualifying exponent. The process involves breaking down each monomial to its basic factors and then extracting the commonalities.

Exponents

Exponents play an essential role in simplifying and understanding algebraic expressions. An exponent, located as a superscript to the right of a variable, indicates how many times to multiply the base number or variable by itself. For instance, x^3 represents x * x * x. This operation condenses what would otherwise be a lengthy multiplication process.

When comparing exponents within monomials to find the GCF, you must look for the smallest exponent of each common variable across the monomials. This smallest exponent represents the highest power you can use as part of the GCF while ensuring that the factor will be common to all monomials in question. Remember, if the variable doesn't appear in all monomials, it cannot be part of the GCF. The concept of exponents is fundamental when reducing algebraic expressions to their simplest forms.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In the context of finding the GCF, we focus on expressions that are exclusively monomials. The GCF of algebraic expressions is the largest expression that divides each of the monomials without leaving a remainder. It's crucial when simplifying fractions with variables or when solving equations where factoring is required.

To determine the GCF of algebraic expressions, one must decompose the expressions into their prime factors or use exponents to express the repeated multiplication of variables. We follow the logic that the GCF is composed of shared variables raised to their lowest exponent present in all terms and any common coefficients. As in the solution provided, this principle guides us to deduce that the GCF of x^2 y, 3x^3y, and 6x^2 is x^2, since 'y' is not present in all terms and 'x' is the only variable shared among the monomials with a minimum exponent of 2.