Factor The Following Expression: 10m 5n - 15

Introduction

Factoring theexpression 10m^5n - 15 is a fundamental skill in algebra that helps simplify equations, solve problems, and reveal hidden patterns in mathematical relationships. In this article we will explore a clear, step‑by‑step method for extracting the greatest common factor (GCF), discuss the scientific reasoning behind why the process works, and answer common questions that arise when learners encounter similar expressions. By the end, you will have a solid framework for tackling any factoring challenge of this nature.

Understanding the Expression

Before we begin factoring, it is essential to recognize the structure of the given term. The expression consists of two parts separated by a subtraction sign:

1. 10m^5n – a monomial that includes a coefficient (10), a variable raised to a power (m^5), and another variable (n).
2. 15 – a constant term with no variables.

The presence of both a variable‑laden term and a pure number means that any common factor must divide both the coefficient and the variable parts. This is the cornerstone of the GCF approach.

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Steps to Factor the Expression

1. Identify the Numerical GCF

• List the prime factors of each coefficient:
  • 10 = 2 × 5
  • 15 = 3 × 5
• The only prime factor they share is 5, so the numerical GCF is 5.

2. Determine the Variable GCF

• The first term contains m^5 and n, while the second term contains no variables.
• Since the second term has no variables, the variable GCF is 1 (i.e., there is no variable factor common to both