Problem 31 - Elementary Algebra

Question

Factor out the GCF. $$ 3 x-18 $$

Step-by-Step Solution

Answer

The factored form is \(3(x - 6)\).

Step 1: Identify the Terms

The expression given is \(3x - 18\). We have two terms here: \(3x\) and \(-18\).

Step 2: Determine the Greatest Common Factor (GCF)

For both terms \(3x\) and \(-18\), we need to find the largest number that divides both coefficients. The coefficients are 3 and 18, and the GCF is 3.

Step 3: Factor the GCF out of Each Term

Divide each term by the GCF (3). For \(3x\), \(\frac{3x}{3} = x\), and for \(-18\), \(\frac{-18}{3} = -6\).

Step 4: Write the Factored Expression

After factoring out the GCF 3 from the expression \(3x - 18\), we are left with \(3(x - 6)\). Therefore, the factored expression is \(3(x - 6)\).

Key Concepts

Greatest Common Factor

Identifying the Greatest Common Factor (GCF) is like finding the biggest piece that both terms in an expression can fit into. For example, in the expression \(3x - 18\), we have two terms: \(3x\) and \(-18\). Each of these terms can be broken down into smaller factors. When we look at the number part of these terms, which is 3 and 18, we want the largest number that divides both perfectly.

• The factor of 3 is 3 itself.
• The factors of 18 are 1, 2, 3, 6, 9, and 18.

Among these numbers, 3 is the greatest factor common to both 3 and 18. By "factoring out the GCF," we mean taking that common piece and seeing what remains of the original terms when divided by this piece. It simplifies how we represent the expression and prepares it for any further algebraic operations if needed.

Expression Simplification

Expression simplification is all about making math expressions easier to work with by reducing them to their simplest form. Simplifying often involves looking for and factoring out elements like the GCF from terms, just like we did with the expression \(3x - 18\).

• This process makes expressions cleaner and more manageable.
• Simplified expressions are easier to interpret and solve in future steps.

In the example, once we found the GCF to be 3, we divided each term by 3. So, \(3x\) became \(x\) and \(-18\) became \(-6\). By doing so, the expression \(3(x - 6)\) reveals the simplified relationship between its terms. The 3 outside the parenthesis shows what factor ties the original terms together, making it easier to work with these parts separately or together.

Algebraic Expressions

Algebraic expressions use variables and constants to represent numeric relationships. They are like sentences in the language of algebra, where numbers and operations are the words and grammar. Expressions like \(3x - 18\) are more complex than basic arithmetic but follow similar rules.

• The term \(3x\) includes a variable "x" with a coefficient 3.
• The term \(-18\) is a constant, which doesn't change.

When factoring or simplifying these expressions, the goal is often to make them more useful for operations such as solving equations or evaluating values. Understanding how expressions are structured enables you to manipulate them effectively, making them friendlier to solve when they appear in equations or other mathematical contexts. Factoring, like we did here, is one way to peel back layers of an expression, exposing its most efficient form.