Problem 12 - Introductory Algebra for College Students

Question

Find the greatest common factor of each list of monomials. $$18 x^{5} y^{4}, 6 x^{6} y^{3}, \text { and } 12 x^{4} y^{5}$$

Step-by-Step Solution

Answer

The greatest common factor of the given monomials is \(6x^{4}y^{3}\).

Step 1: Identifying the GCF of the coefficients

The coefficients of the monomials are 18, 6, and 12. The highest number that divides all of these coefficients evenly is 6 - which becomes the coefficient of the GCF.

Step 2: Identifying the GCF of the powers of x

The powers of \(x\) in the monomials are 5, 6, and 4. The lowest power among these is 4 - which is included in the GCF.

Step 3: Identifying the GCF of the powers of y

The powers of \(y\) in the monomials are 4, 3, and 5. The lowest power among these is 3 - which is included in the GCF.

Step 4: Forming the GCF

Bring together the common coefficient and the powers of \(x\) and \(y\) to form the GCF. So, the GCF is \(6x^{4}y^{3}\).

Key Concepts

Understanding Monomials

Monomials are mathematical expressions that consist of a single term. This term can be a number, a variable, or the product of numbers and variables. Monomials can include constants and variables with whole number exponents. For example, in the original exercise, the monomials given are:

• \(18x^{5}y^{4}\)
• \(6x^{6}y^{3}\)
• \(12x^{4}y^{5}\)

Monomials differ from polynomials as they comprise only one term rather than multiple terms connected by addition or subtraction. This makes it easier to find common factors like the greatest common factor (GCF). Identifying the GCF of monomials involves breaking them down into their individual components and finding the largest common factors shared among them.

The Role of Coefficients

Coefficients in terms like monomials are the numerical parts that are multiplied with the variables. In the exercise, the coefficients are 18, 6, and 12 for each monomial respectively. To find the greatest common factor (GCF), one must identify the greatest number that can divide all coefficients without leaving a remainder.

When determining the GCF of coefficients:

• List the factors of each coefficient
• The common factor with the highest value is chosen as the GCF
• For 18, 6, and 12, the highest common factor is 6

Choosing the largest number that divides all coefficients evenly is crucial for simplifying expressions and solving equations.

Powers of Variables

In monomials, variables can be raised to a certain power, which helps in determining the monomial's degree. The given monomials involve variables \(x\) and \(y\) with different exponents:

• For \(x:\) the powers are 5, 6, and 4
• For \(y:\) the powers are 4, 3, and 5

To find the GCF concerning the powers of variables:

• Identify the lowest power for each variable across all monomials
• The smallest exponent represents the highest power that divides each variable term in the expression
• For \(x\), the lowest power among 5, 6, and 4 is 4
• For \(y\), the lowest power among 4, 3, and 5 is 3

Incorporating these smallest powers helps to determine the most reduced form of common terms when simplifying monomial expressions.