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, demonstrate the step-by-step process, explore different approaches, and delve into the underlying mathematical principles. This guide is designed for students of all levels, from beginners grappling with basic algebra to those seeking a deeper understanding of algebraic manipulation. By the end, you'll not only be able to factorise this specific expression but also possess the tools to tackle similar problems confidently.

Understanding Factorisation

Before we tackle 7y + 21, let's establish a solid foundation in 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. Just as we multiply numbers to get a product, we factorise a product to find its constituent numbers.

For example, the number 12 can be factorised as 2 x 6, 3 x 4, or 2 x 2 x 3. Similarly, algebraic expressions can be factorised into simpler expressions that, when multiplied together, yield the original expression. This process is crucial in simplifying complex expressions, solving equations, and performing other algebraic manipulations.

Step-by-Step Factorisation of 7y + 21

The expression 7y + 21 involves two terms: 7y and 21. To factorise this, we need to identify the greatest common factor (GCF) of these two terms. The GCF is the largest number or expression that divides both terms without leaving a remainder.

Identify the factors of each term:
- 7y: The factors of 7y are 1, 7, y, and 7y.
- 21: The factors of 21 are 1, 3, 7, and 21.
Find the greatest common factor (GCF):

By comparing the factors of both terms, we see that the greatest common factor is 7.
Factor out the GCF:

We can rewrite the expression by factoring out the GCF, 7:

7y + 21 = 7(y + 3)

This is the factorised form of the expression 7y + 21. To verify this, you can expand the factorised form using the distributive property: 7(y + 3) = 7y + 21. This confirms that our factorisation is correct.

Different Approaches to Factorisation

While the method above is straightforward, there are other ways to approach factorisation, especially as expressions become more complex.

Method 1: Identifying the GCF directly: This involves mentally scanning the terms and identifying the largest common factor. This is often the quickest method for simpler expressions like 7y + 21.
Method 2: Prime factorisation: This method involves breaking down each term into its prime factors. For 7y, the prime factors are 7 and y. For 21, the prime factors are 3 and 7. The common prime factor is 7, which is the GCF.
Method 3: Using the distributive property in reverse: This involves thinking about how the distributive property (a(b + c) = ab + ac) can be applied in reverse to factor out a common term. In this case, we observe that both 7y and 21 are divisible by 7, allowing us to factor out 7.

The best method will depend on the complexity of the expression. For simple expressions like 7y + 21, the direct GCF method is usually the most efficient. For more complex expressions, prime factorisation or reversing the distributive property may be more helpful.

Extending the Concept: Factorising More Complex Expressions

The principles used to factorise 7y + 21 apply to a wider range of algebraic expressions. Let's consider some examples:

14x + 28: The GCF is 14. The factorised form is 14(x + 2).
15a - 45b: The GCF is 15. The factorised form is 15(a - 3b).
6x² + 18x: The GCF is 6x. The factorised form is 6x(x + 3). Notice here that we factor out both the numerical and variable parts.
20xy + 15xz + 5x: The GCF is 5x. The factorised form is 5x(4y + 3z + 1).

These examples illustrate that the process remains consistent: identify the GCF and factor it out from each term. As expressions become more complex, you may need to use more sophisticated techniques, but the foundational principles of identifying the greatest common factor remain the same.

The Mathematical Rationale: The Distributive Property

The foundation of factorisation lies in the distributive property of multiplication over addition: a(b + c) = ab + ac. Factorisation is essentially the reverse application of this property. By factoring out the GCF, we are effectively "undoing" the distributive property.

In the case of 7y + 21, we have:

7y + 21 = 7 * y + 7 * 3

Applying the distributive property in reverse, we can write:

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

This shows the clear connection between the distributive property and the process of factorisation. Understanding this relationship provides a deeper understanding of the mathematical justification behind the technique.

Frequently Asked Questions (FAQ)

Q1: What if there is no common factor other than 1?

If there's no common factor other than 1, the expression is already in its simplest form and cannot be further factorised. For example, the expression x + y cannot be factorised.

Q2: Can I factorise expressions with more than two terms?

Yes, you can. The principle is the same: identify the greatest common factor among all the terms and factor it out.

Q3: What if the terms have negative coefficients?

You can factor out a negative GCF. For example, -3x - 6 can be factorised as -3(x + 2).

Q4: How do I factorise expressions with higher powers of variables?

The same principles apply. Consider the expression 4x³ + 8x². The GCF is 4x². The factorised form is 4x²(x + 2).

Q5: Are there other types of factorisation besides finding the GCF?

Yes, there are other more advanced methods such as factoring quadratics (expressions of the form ax² + bx + c) using techniques like difference of squares, perfect square trinomials, and the quadratic formula. These methods are typically introduced at a higher level of algebra.

Conclusion

Factorising algebraic expressions is a fundamental skill in algebra. Understanding the process of identifying the greatest common factor and applying the distributive property in reverse is essential for simplifying expressions, solving equations, and further developing your algebraic proficiency. This article has provided a comprehensive guide to factorising 7y + 21, explored different approaches, and extended the concepts to more complex expressions. With practice and a solid grasp of the underlying principles, you'll be able to confidently factorise a wide variety of algebraic expressions. Remember to always double-check your work by expanding the factorised form to ensure it matches the original expression.