Factorise 7y 21

Factorising 7y + 21: A Comprehensive Guide

This article provides a thorough explanation of how to factorise the algebraic expression 7y + 21. We'll cover the fundamental concepts of factorisation, delve into the step-by-step process, and explore the underlying mathematical principles. Understanding factorisation is crucial for solving algebraic equations, simplifying expressions, and tackling more advanced mathematical concepts. We will also address frequently asked questions and provide examples to solidify your understanding.

Introduction to Factorisation

Factorisation, in its simplest form, is the process of breaking down a mathematical expression into smaller, simpler components – its factors. Think of it like reverse multiplication. If you multiply 2 and 3 to get 6, factorising 6 would be finding those original numbers, 2 and 3. In algebra, we apply the same principle to expressions containing variables (like 'y' in our example). The goal is to identify common factors within the expression and rewrite it as a product of those factors.

Step-by-Step Factorisation of 7y + 21

Let's tackle the expression 7y + 21. The key is to identify the greatest common factor (GCF) shared by both terms, 7y and 21.

1. Identify the factors of each term:

   • 7y: The factors are 7 and y.
   • 21: The factors are 1, 3, 7, and 21.

2. Find the greatest common factor (GCF):

   By examining the factors of both terms, we see that the largest number that divides both 7y and 21 evenly is 7. Therefore, the GCF is 7.

3. Factor out the GCF:

   We now rewrite the expression by factoring out the GCF (7) from both terms:

   7y + 21 = 7(y + 3)

This is the factorised form of 7y + 21. We've successfully broken down the original expression into a product of two factors: 7 and (y + 3).

Understanding the Process: A Deeper Dive

The process we followed relies on the distributive property of multiplication. The distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

In our case, a = 7, b = y, and c = 3. So, applying the distributive property in reverse, we went from 7y + 21 (ab + ac) to 7(y + 3) (a(b + c)).

Expanding the Factorised Expression to Verify

To verify our factorisation is correct, we can expand the factorised expression using the distributive property:

7(y + 3) = 7 * y + 7 * 3 = 7y + 21

This confirms that our factorisation is accurate, as expanding the factorised form gives us the original expression.

More Complex Examples of Factorisation

While 7y + 21 is a relatively straightforward example, let's explore some more complex scenarios to further illustrate the principles of factorisation.

Example 1: Factorising 15x² + 30x

1. Identify the factors of each term:

   • 15x²: Factors are 1, 3, 5, 15, x, x², and various combinations.
   • 30x: Factors are 1, 2, 3, 5, 6, 10, 15, 30, x, and various combinations.

2. Find the greatest common factor (GCF):

   The GCF of 15x² and 30x is 15x.

3. Factor out the GCF:

   15x² + 30x = 15x(x + 2)

Example 2: Factorising 4a²b - 8ab² + 12ab

1. Identify the factors of each term:

   • 4a²b: Factors include 1, 2, 4, a, a², b, and combinations.
   • 8ab²: Factors include 1, 2, 4, 8, a, b, b², and combinations.
   • 12ab: Factors include 1, 2, 3, 4, 6, 12, a, b, and combinations.

2. Find the greatest common factor (GCF):

   The GCF of 4a²b, 8ab², and 12ab is 4ab.

3. Factor out the GCF:

   4a²b - 8ab² + 12ab = 4ab(a - 2b + 3)

Factorisation and Solving Equations

Factorisation is a crucial tool in solving algebraic equations. Consider a quadratic equation like:

x² + 5x + 6 = 0

To solve this, we first factorise the quadratic expression:

(x + 2)(x + 3) = 0

This equation is true if either (x + 2) = 0 or (x + 3) = 0. Solving these simpler equations gives us the solutions x = -2 and x = -3.

Frequently Asked Questions (FAQ)

• What happens if there's no common factor? If the terms in an expression share no common factors other than 1, then the expression is considered already in its simplest factorised form.

• Can I factorise expressions with more than two terms? Yes, the principles of finding the GCF and factoring it out apply to expressions with any number of terms.

• What if the expression involves negative numbers? The GCF can be negative. For example, in -6x - 12, the GCF is -6, leading to the factorised form -6(x + 2).

• How do I factorise more complex expressions? More advanced techniques exist for factorising complex polynomials, including techniques like grouping and using the quadratic formula, which are topics for more advanced study.

Conclusion

Factorising algebraic expressions, even seemingly simple ones like 7y + 21, is a fundamental skill in algebra. Understanding the process of identifying the greatest common factor and applying the distributive property allows you to simplify expressions, solve equations, and build a stronger foundation for more advanced mathematical concepts. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems and confidently navigate your mathematical journey. Remember, practice makes perfect! Continue working through various examples, and don't hesitate to review the steps involved whenever you need to. With consistent effort, you'll develop a solid understanding of factorisation and its applications.