Problem 43 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ 18 m-9 $$

Step-by-Step Solution

Answer

The factored expression is \(9(2m - 1)\).

Step 1: Identify the Terms

The given expression is \(18m - 9\). There are two terms here: \(18m\) and \(-9\).

Step 2: Determine the GCF

Find the greatest common factor (GCF) of the coefficients 18 and 9. The factors of 18 are 1, 2, 3, 6, 9, 18, and the factors of 9 are 1, 3, 9. The greatest common factor is 9.

Step 3: Factor Out the GCF

Divide each term in the expression by the GCF, which is 9. This gives us: \( \frac{18m}{9} = 2m \) and \( \frac{-9}{9} = -1 \).

Step 4: Write the Factored Expression

Output the expression as \(9(2m - 1)\). This is the expression with the GCF factored out.

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a fundamental concept in algebra that helps simplify expressions. When dealing with numbers, the GCF is the largest number that divides two or more numbers without leaving a remainder.
In the context of algebraic expressions, the GCF pertains to the coefficients of terms involved. To find the GCF, you follow these steps:

• List out all the factors of each number involved. For example, for 18, the factors are 1, 2, 3, 6, 9, 18.
• Do the same for the other number in the expression, such as 9, resulting in factors of 1, 3, 9.
• Identify the largest number common to both lists. For our example, this is 9.

Once the GCF is identified, it can be factored out of the expression, simplifying it for further work.

Factored Expression

A factored expression is a way of writing an expression as a product of its factors. This simplification makes it easier to handle complex algebraic equations.
Let's revisit our example, where we have the expression \(18m - 9\). After determining the GCF, which is 9, we factor it out from the expression by dividing each term by this GCF.

• This means dividing \(18m\) by 9 to get \(2m\).
• Then, divide \(-9\) by 9 to get \(-1\).

The factored expression becomes \(9(2m - 1)\). This form shows that the GCF is a common factor of all terms in the original expression, simplifying it effectively for further calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operational symbols such as \(+\), \(-\), \(\times\), and \(\div\). They can be as simple as \(18m - 9\) or more complex, involving multiple variables and operations.
Understanding how to manipulate these expressions is crucial in algebra. A fundamental skill is identifying parts of the expression, such as terms, coefficients, and variables. In \(18m - 9\):

• There are two terms: \(18m\) and \(-9\).
• "m" is a variable that can represent any number.
• The coefficients of the terms are 18 and -9, respectively.

By factoring out the GCF, we simplify the expression, making it a powerful tool for solving equations or further algebraic manipulation.