Problem 21 - Algebra for College Students

Question

Factor completely. $$6 x+3 y$$

Step-by-Step Solution

Answer

The expression is factored as \(3(2x + y)\).

Step 1: Identify Common Factors

First, observe the coefficients and terms in the expression \(6x + 3y\). Identify the greatest common factor (GCF) for the numbers 6 and 3, which is 3, because 3 is the largest number that divides both coefficients evenly.

Step 2: Factor Out the GCF

Divide each term by the GCF identified (3): \( \frac{6x}{3} = 2x \) and \( \frac{3y}{3} = y \). This results in the factored expression \(3(2x + y)\).

Step 3: Write the Final Factored Expression

Write down the expression with the factored terms: The complete factorization of \(6x + 3y\) is \(3(2x + y)\).

Key Concepts

Greatest Common Factor

When you are trying to factor expressions like \(6x + 3y\), the greatest common factor (GCF) is your best friend. The GCF is the largest number that can evenly divide each coefficient of the terms in the expression. For the coefficients 6 and 3 in our example, the GCF is 3. This means 3 can divide both numbers without leaving a remainder.
To find the GCF, list the factors of each number. For 6, the factors are 1, 2, 3, and 6. For 3, the factors are 1 and 3. The greatest number shared between them is 3, making it the GCF.
Once identified, this GCF can be used to simplify or "factor out" the expression further by dividing each term by 3, leading to an easier-to-manage algebraic expression.

Distributive Property

The distributive property plays a key role when factoring expressions. It states that \(a(b + c) = ab + ac\). In reverse, this means you can factor an expression by identifying common elements or terms.
In the example \(6x + 3y\), once we identify the GCF as 3, we apply the distributive property. Each term, 6x and 3y, can be divided by the GCF 3.

• Divide 6x by 3 to get 2x
• Divide 3y by 3 to get y

This transforms our expression into \(3(2x + y)\).
The distributive property thus allows us to rewrite the expression in a factored form, simplifying the expression to reveal its basic components.

Algebraic Expressions

An algebraic expression refers to a mathematical phrase that can contain numbers, variables (like \(x\) or \(y\)), and operations (like +, -, *, and /). Factoring expresses the expression in simpler, reduced terms, making it easier to solve or understand.
For instance, in the original expression \(6x + 3y\), recognizing parts of the expression can be important. Look at the coefficients, like 6 and 3, and the variables, \(x\) and \(y\). The first step in simplifying algebraic expressions is often to find the GCF of the coefficients.

• Identify any shared factors across all components, such as 3 in this case.
• Divide throughout by this GCF to express the simpler form: \(3(2x + y)\).

Understanding how algebraic expressions work enables us to manipulate and solve them more efficiently, making math problems easier to tackle.