Problem 11 - Prealgebra

Question

Factor each expression. $$3 n+9$$

Step-by-Step Solution

Answer

The factored expression is \(3(n + 3)\).

Step 1: Identify the Common Factor

The first step in factoring any expression is to identify the greatest common factor (GCF) of the terms. In the expression \(3n + 9\), observe the coefficients and terms: \(3n\) and \(9\). Both terms are divisible by \(3\). Thus, the GCF is \(3\).

Step 2: Apply the Distributive Property

Use the distributive property to factor out the GCF. Divide each term by the GCF and rewrite the expression. \[3n + 9 = 3(n + 3)\] Here, we factor out \(3\) from the expression, leaving us with \(n + 3\) inside the parenthesis.

Step 3: Check Your Work

Finally, it is crucial to check if the factoring is done correctly by expanding the factored expression. Use distribution: \[3(n + 3) = 3n + 9\] This matches the original expression, confirming the factoring is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides each term in an expression without leaving a remainder. When we look at the expression \(3n + 9\), both terms, \(3n\) and \(9\), need to be considered.

• For the term \(3n\): The coefficient is \(3\).
• For the term \(9\): The number itself is \(9\).
• Both \(3\) and \(9\) are divisible by \(3\).

Thus, the GCF of \(3n\) and \(9\) is \(3\). Factoring makes expressions simpler by showing the building blocks of the terms, which is especially helpful in solving algebraic equations.

Distributive Property

The distributive property allows us to break down expressions into simpler parts by taking a common factor and distributing it across terms.When we apply it to the expression \(3n + 9\):

• We factor out the GCF, which is \(3\).
• Express the original terms as products of the GCF.

So, \(3n + 9\) can be written as \(3(n + 3)\). Hence, \(3\) is distributed across \(n + 3\), simplifying the expression and making it easier to work with in more complex equations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition and multiplication. They are the building blocks of algebra and essential in expressing real-life problems mathematically.In the expression \(3n + 9\):

• \(3n\): This is a term with a coefficient \(3\) and a variable \(n\).
• \(+ 9\): A constant term added to \(3n\).

When factoring, we see that these expressions can be decomposed into simpler terms using the concepts of the greatest common factor and the distributive property to enhance understanding and solve for unknown variables efficiently. Factoring helps in transforming complex expressions, which forms the cornerstone of many algebraic processes.