Problem 4 - College Algebra with Corequisite Support

Question

For the following exercises, find the greatest common factor. \(14 x+4 x y-18 x y^{2}\)

Step-by-Step Solution

Answer

The greatest common factor is \(2x\).

Step 1: Identify the Common Factor

Look for the common factor in all terms of the expression. Each term is preceded by a numerical coefficient and contains the variable 'x'. The terms are: \(14x, 4xy, -18xy^2\). Compare the coefficients (14, 4, -18) and the variables (x, xy, xy^2) to find common factors.

Step 2: Find the GCD of Numerical Coefficients

Now find the greatest common divisor (GCD) of the coefficients 14, 4, and 18. The prime factors are: 14 = 2 * 7, 4 = 2^2, 18 = 2 * 3^2. The common factor from the numerical coefficients is 2.

Step 3: Consider the Variable Part

Analyze the variables in each term: \(x, xy, xy^2\). All terms share the factor \(x\). Thus, the common variable factor is \(x\).

Step 4: Combine Factors

Combine the common factors from Step 2 (2) and Step 3 (x). Thus, the greatest common factor (GCF) of the expression is the product of these common factors, which is \(2x\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. These expressions can include terms that consist of both numerical coefficients and variables. For example, in the expression \(14x + 4xy - 18xy^2\), each term is separate and includes a number and variable(s), such as \(14x\), \(4xy\), and \(-18xy^2\).
Breaking down these terms:

• \(14x\) has the number 14, called the numerical coefficient, and the variable \(x\).
• \(4xy\) has 4 as the numerical coefficient, with variables \(x\) and \(y\).
• \(-18xy^2\) consists of -18 (numerical coefficient) and variables \(x\) and \(y^2\).

Understanding these expressions is fundamental as they are the building blocks of algebra. Recognizing coefficients and variable parts helps in identifying common factors in complex polynomials.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of its factors. This process can simplify algebraic expressions and solve equations efficiently. When we want to find the greatest common factor (GCF) of a polynomial like \(14x + 4xy - 18xy^2\), we look to factor out the largest common term from each part.
Factorizing helps break down expressions into simpler components:

• Find common numerical factors like 2 in the coefficients 14, 4, and 18.
• Identify the common variable factors; in this case, \(x\) is common in all terms.

Ultimately, factorization allows for expressions to be written in a simpler form that reveals their structure, leading to easier manipulation and solution finding.

Numerical Coefficients

Numerical coefficients are the numbers placed in front of variables in algebraic terms. They play a crucial role in simplifying expressions and finding common factors. In the expression \(14x + 4xy - 18xy^2\), the coefficients are 14, 4, and -18.
When looking for the greatest common factor, we assess these coefficients first by determining their greatest common divisor (GCD). Here's the breakdown:

• 14 factors as \(2 \times 7\).
• 4 factors as \(2^2\).
• 18 factors as \(2 \times 3^2\).

The common factor is the number 2. Understanding and identifying numerical coefficients helps in efficiently finding GCDs and simplifying expressions by taking out the greatest common factor.