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The greatest common factor (GCF) is the largest number that can evenly divide all the terms in an expression.
Finding the GCF is crucial in simplifying expressions because it allows us to factor out this common value, making the expression simpler and often easier to work with.

For the expression given in the exercise, $$9p-3$$:

• The number 9 and 3 share the common factor of 3.
• So, the GCF of 9p and 3 is 3.

Identifying the GCF is always the first step in factoring any algebraic expression. To find it, you can:

• List out the factors of each term.
• Choose the largest factor that both terms share.

This makes breaking down the expression into simpler terms much smoother.

Equivalent expressions are different expressions that represent the same value.
When you factor an expression, you create an equivalent expression that is easier to work with.
Both the original and the factored forms represent the same mathematical relationship. Let's see this concept in action using the given expression:$$9p - 3:$$

• Original expression: 9p - 3
• Factored expression: 3(3p - 1)

These two forms are equivalent because if you distribute the 3 in the factored form: \ \rightarrow 3(3p) - 3(1)
You will get back the original expression: \ \rightarrow 9p - 3
This confirms they are indeed equivalent. Understanding this concept helps to know why and how we use factoring as a technique to simplify equations.

Factoring is a technique used to simplify algebraic expressions by finding a product of simpler factors.
This technique breaks down complex expressions into multiplied factors, which are easier to handle in operations like solving equations or simplifying. There are several techniques, but in our exercise, we specifically looked at factoring out the greatest common factor (GCF).
Here's how you can do it:

• Identify the GCF of all terms in the expression.
• Rewrite each term as a product of the GCF and another term.
• Factor out the GCF.

Let’s go through the steps as shown in the exercise for 9p-3: 1. Identify GCF: The GCF of 9p and 3 is 3. 2. Rewrite each term: 9p = 3 * 3p and 3 = 3 * 1. 3. Factor out 3: 3(3p - 1)
Following these steps ensures you end up with a factored expression that is correct and equivalent to the original.