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Chapter 4: Problem 53

Factor each expression. $$5 n-10 m+25$$

Short Answer

Expert verified
The factored expression is \(5(n - 2m + 5)\).

Step by step solution

01

Identify the common factor

First, look for the greatest common factor (GCF) in the terms of the expression \(5n - 10m + 25\). Notice that each coefficient (5, -10, 25) is divisible by 5. Thus, the GCF is 5.
02

Factor out the GCF

Use the GCF to factor the expression. Divide each term by 5 and express the original equation in a new form: \[5n - 10m + 25 = 5(n) - 5(2m) + 5(5)\]. Now, factor out the 5: \[5(n - 2m + 5)\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
The greatest common factor (GCF) is the largest number that divides all numbers in a set without leaving a remainder. This concept is a key tool in simplifying expressions. In our exercise, we need to find the GCF of the terms 5n, -10m, and 25. Each coefficient in this expression, namely 5, -10, and 25, is divisible by 5.
Doing this reduction, we identify the GCF as 5. Recognizing this commonality allows us to factor the expression efficiently and makes any further algebraic manipulation easier.
Expression Simplification
Simplifying expressions is a fundamental skill in algebra that can help make problems more manageable. After identifying the GCF, the next step is to factor it out from the expression. This process involves dividing each term by the GCF found in the previous step.
So for the expression on hand, we divide each term:
  • 5n becomes n
  • -10m becomes -2m
  • 25 becomes 5
This step rewrites the original equation into a simpler form, which is now expressed as 5(n - 2m + 5).
Simplifying expressions in this manner helps reveal its underlying structure, facilitating further work with the equation.
Prealgebra Techniques
Prealgebra is the stepping stone to more advanced algebraic concepts. It involves foundational techniques such as identifying the greatest common factor and simplifying expressions, as illustrated in our exercise.
These skills are vital for solving basic algebra problems and serve as a useful prelude to tackling more complex equations later on. Prealgebra provides the necessary frameworks, ensuring that students have a solid grasp of concepts that allow them to progress through mathematics with confidence.
  • Practice identifying common factors in different sets of numbers.
  • Regularly simplify various expressions as an exercise.
By mastering these techniques, students build a strong mathematical foundation that supports future learning in algebra and beyond.

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