Question

$$6 x y z+12 x^{2} y^{2} z$$

Short answer

The simplified expression is \(6xyz(1 + 2xy)\).

Step-by-step solution

Factor out the common factors

Identify the common factors in both terms. Here, both terms have the common factors of \(x, y, z\), and also a numerical factor that can be factored out. The greatest common factor (GCF) that can be factored out from both terms is \(6xyz\).

Apply the distributive property

Factor out the GCF from both terms using the distributive property, also known as 'factoring out'. This will split the original expression into the GCF multiplied by the sum of what remains of each term.

Write down the simplified expression

After factoring out the GCF, rewrite the original expression as the product of the GCF and the remaining terms.

Key concepts

Greatest Common Factor (GCF)

When factoring algebraic expressions, the Greatest Common Factor, or GCF, is the first and most crucial step in simplifying the expression. The GCF is the largest factor that divides two or more numbers or terms. In the context of algebra, it includes not only numerical values but also the common variables and their lowest powers.

To identify the GCF in an algebraic expression, look for the highest number that can evenly divide the coefficients, and then for each variable present, use the lowest power of that variable found in the terms. For example, in the expression \(6xyz + 12x^2y^2z\), both terms have \(x\text{, } y\text{, and } z\) as variables. The numerical GCF is \(6\) as it is the largest number that divides the coefficients 6 and 12. After identifying the numerical GCF and the common variables, we combine them to find the overall GCF, which is \(6xyz\) in this case.

Recognizing the GCF allows us to factor expressions cleanly and effectively, setting the stage for further simplification.

• Identify numerical GCF
• Identify common variables with the lowest power
• Combine numerical and variable factors for the overall GCF

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms within parentheses. When we have found the GCF of an algebraic expression, we can use the distributive property in reverse, which is often referred to as 'factoring out' the GCF.

Applying the distributive property, we multiply the GCF across each term of the expression. To reverse this process, we look at each term of the expression, identify the GCF, and divide each term by this GCF; we effectively 'distribute' the division. The original expression is then rewritten as the GCF multiplied by a new expression created by these divided terms.

For instance, with our example, \(6xyz + 12x^2y^2z\), we factor out the GCF, \(6xyz\), and use the distributive property to get \(6xyz(1 + 2xy)\). Each term within the parentheses is a result of dividing the original terms by \(6xyz\).

Remembering to use the distributive property in this way is essential for correctly factoring expressions.

• Use distributive property in reverse to factor out GCF
• Divide each term by the GCF
• Rewrite the expression as GCF multiplied by quotient terms

Simplifying Expressions

After identifying the GCF and utilizing the distributive property, the third step is simplifying expressions. This process involves rewriting the factored expression in a simpler form without changing its value. The goal is to make the expression easier to understand and to work with in subsequent mathematical operations.

In our example, once we have factored out \(6xyz\), the expression simplifies to \(6xyz(1 + 2xy)\). Now, this new expression represents the same quantity as the original but is organized in a more straightforward manner. It becomes evident that any further operations, like addition or subtraction, will only need to be carried out within the parentheses.

It's vital to ensure that after simplifying, there are no common factors left other than 1. Simplifying expressions reduces them to their most manageable form and is critical in solving equations and understanding mathematical relationships.

Simplification may involve:

• Combining like terms
• Reducing fractions
• Cancelling common factors within numerators and denominators