Problem 129 - Algebra and Trigonometry

Question

What does it mean to factor completely?

Step-by-Step Solution

Answer

To factor completely means to express a number or expression as a product of prime numbers (for numbers) or irreducible polynomials (for expressions). For instance, the number 12 can be factored completely into 2x2x3.

Step 1: Define Factoring

Factoring in mathematics means expressing a given number or expression as a product of two or more other numbers or expressions. These numbers or expressions are called the 'factors' of the original number or expression.

Step 2: Understand the Concept of Prime Numbers

Prime numbers are those numbers that have only two factors: 1 and the number itself. They cannot be expressed as the product of other numbers. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

Step 3: Define 'Factoring Completely'

When a number or expression is expressed as the product of prime numbers (for numbers) or irreducible polynomials (for expressions), it is said to be 'factored completely'.

Step 4: Apply the Concept Through Examples

For instance, if asked to factor the number 12 completely, the process would be: 12 can be factored into 3x4, but it's not factored completely because 4 isn't a prime number. Factoring 4 further gives 2x2. So, the completely factored form of 12 is 2x2x3, which are all prime numbers.