Problem 84 - Elementary Algebra

Question

Either factor out the greatest common factor or factor by grouping. $$ 8 a b+2 c d+6 c f $$

Step-by-Step Solution

Answer

2(4ab + cd + 3cf)

Step 1: Identify Components

Look at each term in the expression: \(8ab\), \(2cd\), and \(6cf\). Determine if there is a common factor among these terms.

Step 2: Find the Greatest Common Factor (GCF)

Examine the coefficients (8, 2, and 6) first. The greatest common factor among them is 2. Now examine the variables: there are no common variables in all terms. So, the GCF is 2.

Step 3: Factor Out the GCF

Factor out 2 from each term: \[ 8ab + 2cd + 6cf = 2(4ab + cd + 3cf) \] This is your factored expression.

Key Concepts

greatest common factor

To understand factoring, it's crucial to know about the Greatest Common Factor (GCF). The GCF is the largest number that can evenly divide all the numbers in a set. When factoring algebraic expressions, this concept helps in simplifying expressions. Identify the GCF by:

• Listing the factors of each term's coefficient.
• Choosing the largest factor common to each list.

In our example, the coefficients are 8, 2, and 6. Their GCF is 2, as 2 is the highest number that can divide 8, 2, and 6 evenly.

factoring expressions

Factoring expressions means breaking them down into simpler parts (or factors) that, when multiplied together, give the original expression. After finding the GCF, you factor it out of the expression.
In the example:

8ab + 2cd + 6cf

Here’s how to factor it:
· Identify the GCF, which is 2.
· Divide each term by the GCF (every term’s coefficient).
· Re-write the expression with the GCF factored out:
\[ 2(4ab + cd + 3cf) \]
This simplifies the expression, making it easier to work with in equations.

algebraic terms

Understanding algebraic terms is integral for factoring. An algebraic term is a part of a mathematical expression separated by + or -. Each term can include:

• A coefficient (a number).
• Variables (like a, b, c).

In the example:
\(8ab + 2cd + 6cf\)
Each term has a coefficient (8, 2, 6) and variables (ab, cd, cf). Recognizing and separating these will help you in identifying common factors and simplifying expressions. For example, in our expression, 8, 2, and 6 are the coefficients that are handled first while factoring.