Problem 9 - College Algebra

Question

Factor out the greatest common factor. $$ x^{2}(x-3)+12(x-3) $$

Step-by-Step Solution

Answer

The expression \(x^{2}(x-3)+12(x-3)\) after factoring out the greatest common factor is \((x-3)(x^{2}+12)\).

Step 1: Identify the common factor

Observe the expression \(x^{2}(x-3)+12(x-3)\). We can see that (x-3) is the common factor in each term.

Step 2: Factor out the common factor

Apply the distributive property, which reverses the multiplication of a term. So, factor out (x-3) to get \((x-3)(x^{2}+12)\).

Key Concepts

Greatest Common Factor

Factoring is a process where we find values that can divide each term of an expression evenly. In this exercise, we aim to find and remove the greatest common factor (also called the GCF), which simplifies the expression. The GCF is the largest value that divides all terms completely, helping to make expressions easier to handle.
For the expression \(x^{2}(x-3)+12(x-3)\), you can see that \((x-3)\) is a part of both terms. This means \((x-3)\) is a common factor, and because it is present in all terms, it serves as the expression's GCF in the context of these terms. By identifying the GCF, we can simplify expressions more efficiently.

Distributive Property

The distributive property is a central concept in algebra which allows us to multiply a single term across terms inside a parenthesis. It works like a reverse process in factoring.
In the expression \(x^{2}(x-3)+12(x-3)\), by applying the distributive property in reverse, we factor out the \((x-3)\) from each term. This means \((x-3)\) is multiplied across the other terms within the parentheses. The process forms an equivalent expression:

• Original: \(x^{2}(x-3) + 12(x-3)\)
• Factored: \((x-3)(x^2 + 12)\)

This shows how the distributive property helps us understand the structure of expressions and makes large equations more manageable.

Common Factor

Finding a common factor is key in algebraic manipulation and simplification. A common factor is any number or expression that divides each term of an equation evenly. It sounds simple but makes a big impact on handling algebraic problems.
In our exercise, \((x-3)\) is the common factor in the terms \(x^{2}(x-3)\) and \(12(x-3)\). Each of these terms contains \((x-3)\), indicating it can be factored out. This process involves grouping the common elements to stand outside a set of parentheses, simplifying the expression.
Recognizing and factoring the common factors helps in reshaping the equation, usually leading to easier solutions, which is a helpful skill in both basic and advanced algebraic problems.