Problem 17 - Elementary Algebra

Question

Determine the GCF of all the terms. $$ 12 x 2 y, 60 x y 3 $$

Step-by-Step Solution

Answer

The GCF of the terms is \(12xy\).

Step 1: Identify the Numerical GCF

Look at the numerical coefficients of each term: 12 and 60. List the factors of these numbers.\[12: 1, 2, 3, 4, 6, 12\] \[60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\] The greatest common factor (GCF) from these lists is 12.

Step 2: Analyze the Variable Parts

For the variables, look for each variable’s lowest degree since that will be the common factor. In 12, we have \(x^2y\) and in 60, we have \(xy^3\). Compare the degrees of each variable across terms: \(x^2\) and \(x\) gives us \(x\), \(y\) and \(y^3\) gives us \(y\).

Step 3: Compile the Overall GCF

Combine the numerical GCF and the variable parts identified. From Step 1, the numerical GCF is 12. From Step 2, the variables are \(xy\). Therefore, the overall GCF of the terms \(12x^2y\) and \(60xy^3\) is \(12xy\).

Key Concepts

Numerical Coefficients

Numerical coefficients are the number part of the terms in an expression or polynomial. In our example, the terms are \(12x^2y\) and \(60xy^3\), where the numerical coefficients are 12 and 60. These coefficients are crucial because they determine the size of each term. To find the Greatest Common Factor (GCF) among numerical coefficients, list all the factors of each number.

• Factors of 12 are: 1, 2, 3, 4, 6, 12.
• Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

The largest factor common to both numbers is 12. Therefore, the numerical GCF is 12. Finding the numerical GCF is a critical step in simplifying polynomial expressions, as it allows us to manage the coefficients effectively.

Variable Exponents

Variable exponents refer to the power to which variables are raised in algebraic terms. In the terms \(12x^2y\) and \(60xy^3\), pay attention to both \(x\) and \(y\) variables. Each variable has an exponent indicating how many times to multiply the variable by itself.

• For \(x\), the exponents are 2 in \(x^2y\) and 1 in \(xy^3\).
• For \(y\), the exponents are 1 in \(x^2y\) and 3 in \(xy^3\).

To find the common factor in terms of variables, select the lowest exponent for each variable. For \(x\), take \(x^1\) or simply \(x\). For \(y\), take \(y^1\) or simply \(y\). The variable part of the GCF is therefore \(xy\). Understanding variable exponents is essential, as they determine the behavior and characteristics of the terms in a polynomial.

Factorization

Factorization is the process of breaking down expressions into their simplest components. It's like finding what numbers and variables multiply together to create the original expression.
For example, start with the term \(12x^2y\). Factor it into its numerical and variable parts as \(12 \times x^2 \times y\). Similarly, factor \(60xy^3\) into \(60 \times x \times y^3\). Once factored, it is easier to identify common parts. The GCF for both terms involves combining the numerical and variable common factors found earlier.

• The numerical GCF is 12.
• The variable GCF is \(xy\).

So, the overall GCF for simplifying \(12x^2y\) and \(60xy^3\) after factorization is \(12xy\). This process helps in simplifying expressions, making calculations easier, and gaining insights into the structure of polynomial expressions.