Problem 129 - College Algebra

Question

What does it mean to factor completely?

Step-by-Step Solution

Answer

Factoring completely involves breaking down a number or expression into its simplest 'building blocks', usually prime numbers or variables raised to the power of 1.

Step 1: Understanding Factoring

Factoring is a process in mathematics which involves breaking down numbers or expressions into products of other numbers or expressions. For example, the number 10 can be factored into 2 and 5, because 2 * 5 = 10.

Step 2: Factoring Completely

Factoring completely means breaking down a number or expression into its simplest terms, which cannot be factored any further. For an equation, for example, this means breaking it down until each term is a prime number or a variable with a power of 1. For example, factoring the number 18 completely would result in 2 * 3 * 3, because these are prime numbers and cannot be factored any further.

Step 3: Factoring and Algebra

Factoring is also important in algebra, as one often wants to simplify expressions or solve equations. Factors are the 'building blocks' of these processes. For example, the expression \(x^2 - 5x + 6\) can be factored completely as \((x - 2)(x - 3)\), and these simpler expressions can be used to find solutions for x.