What is a factor? Give an examl

A **factor** is a number or algebraic expression that divides another number or expression evenly, meaning there is no remainder left over. Factors are essential in various areas of mathematics, including arithmetic, algebra, and number theory. ### **Example in Arithmetic:** Consider the number **12**. To find its factors, identify all whole numbers that can divide 12 without leaving a remainder. - **1 × 12 = 12** - **2 × 6 = 12** - **3 × 4 = 12** So, the factors of 12 are **1, 2, 3, 4, 6,** and **12**. ### **Example in Algebra:** In algebra, factoring involves breaking down an expression into simpler components (factors) that, when multiplied together, give the original expression. **Example Expression:** \( x^2 - 5x + 6 \) **Factoring Process:** 1. **Find two numbers** that multiply to **+6** (the constant term) and add up to **-5** (the coefficient of the middle term). 2. The numbers **-2** and **-3** fit because: - \( (-2) \times (-3) = 6 \) - \( (-2) + (-3) = -5 \) **Factored Form:** \( (x - 2)(x - 3) \) Here, **\( (x - 2) \)** and **\( (x - 3) \)** are factors of the quadratic expression \( x^2 - 5x + 6 \). ### **Key Points:** - **Factors in Numbers:** Identify integers that divide the given number exactly. - **Factors in Algebra:** Break down expressions into product of simpler expressions. Understanding factors is fundamental in simplifying mathematical problems, solving equations, and analyzing numerical relationships.