Problem 129 - College Algebra Essentials

Question

What does it mean to factor completely?

Step-by-Step Solution

Answer

Factoring completely means breaking down an algebraic expression into its simplest forms, where no further factoring is possible. It's simplifying an expression to its smallest building blocks, or factors, which are typically prime numbers and variables.

Step 1: Understanding Factoring

Factoring refers to breaking an algebraic expression down into a product of simpler terms. These 'factors' when multiplied together will result in the original expression.

Step 2: Define Complete Factoring

When we talk about factoring 'completely', it means that we continue the process of factoring until we cannot factor the expression any further. In other words, an expression is said to be factored completely if no common factors other than 1 can be found within the terms of an expression.

Step 3: Example

As an example, consider the expression \(36x^2y\). We can factor this as \(6 \cdot 6 \cdot x \cdot x \cdot y\), but that's not completely factored. Factoring completely would look like this: \(2 \cdot 2 \cdot 3 \cdot 3 \cdot x \cdot x \cdot y\). Now, every factor is a prime number, or a letter, and we have factored completely.