Factorise 5x 15

Factorising 5x + 15: A Comprehensive Guide

This article provides a comprehensive guide to factorising the algebraic expression 5x + 15. We'll explore the concept of factorisation, demonstrate the steps involved in factorising this specific expression, delve into the underlying mathematical principles, and address frequently asked questions. Understanding factorisation is crucial for various mathematical operations and problem-solving in algebra and beyond. This guide is designed for learners of all levels, from beginners grappling with the basics to those seeking a deeper understanding.

Introduction to Factorisation

Factorisation, in its simplest form, is the process of breaking down a mathematical expression into smaller components, or factors, that when multiplied together, produce the original expression. Think of it like reverse multiplication. Just as we can multiply numbers to get a product, factorisation allows us to decompose a number or expression into its constituent parts. This skill is fundamental in algebra, calculus, and numerous other mathematical fields.

In the context of algebraic expressions, factorisation involves identifying common factors among the terms and expressing the expression as a product of these factors. This simplifies the expression and allows for easier manipulation and problem-solving in more complex equations.

Step-by-Step Factorisation of 5x + 15

Let's break down the factorisation of 5x + 15 step-by-step:

1. Identify the terms: The expression 5x + 15 consists of two terms: 5x and 15.

2. Find the greatest common factor (GCF): This is the most crucial step. We need to identify the largest number or variable that divides both terms without leaving a remainder. Let's examine the numerical coefficients (the numbers in front of the variables) and the variables themselves:

   • Numerical coefficients: 5 and 15. The greatest common factor of 5 and 15 is 5.
   • Variables: The first term contains 'x', while the second term does not. Therefore, there's no common variable factor.

3. Factor out the GCF: Now, we extract the GCF (which is 5) from both terms:

   5x + 15 = 5(x) + 5(3)

4. Rewrite the expression: Since 5 is a common factor, we can rewrite the expression as a product of the GCF and the remaining terms within parentheses:

   5x + 15 = 5(x + 3)

Therefore, the factorised form of 5x + 15 is 5(x + 3). This means that 5 and (x + 3) are the factors of the original expression. If you were to expand this factorised form (using the distributive property), you would obtain the original expression 5x + 15.

Deeper Dive into the Mathematical Principles

The process of factorisation relies on several fundamental mathematical principles:

• Distributive Property: This property states that a(b + c) = ab + ac. This is the foundation of factorisation. We essentially reverse this property to extract the common factor.

• Greatest Common Divisor (GCD): Finding the GCD is crucial for efficient factorisation. The GCD of a set of numbers is the largest number that divides all the numbers in the set without leaving a remainder. Algorithms like the Euclidean algorithm can be used to find the GCD of larger numbers.

• Prime Factorisation: For more complex expressions, prime factorisation of the coefficients can be helpful. Prime factorisation involves expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves). This helps in identifying the GCF more easily. For instance, the prime factorisation of 15 is 3 x 5, clearly showing that 5 is a factor.

Factorisation of More Complex Expressions

While 5x + 15 is a relatively simple example, the principles applied here extend to more complex algebraic expressions. Consider the following examples:

• 10x² + 20x: The GCF here is 10x. Therefore, the factorised form is 10x(x + 2).

• 3x³ + 6x² + 9x: The GCF is 3x. The factorised form is 3x(x² + 2x + 3).

• Expressions with more than two terms: The same principle applies. Find the GCF of all terms and factor it out. For example, in 4x²y + 8xy² + 12xyz, the GCF is 4xy, resulting in 4xy(x + 2y + 3z).

• Difference of squares: A special case involves expressions of the form a² - b², which factors as (a + b)(a - b). For example, x² - 9 = (x + 3)(x - 3).

• Trinomials: Trinomials (expressions with three terms) often require more advanced techniques, such as factoring by grouping or using the quadratic formula.

Frequently Asked Questions (FAQ)

Q: What if there is no common factor?

A: If there's no common factor among the terms, the expression is already in its simplest form, and it cannot be further factorised. For example, x + 7 cannot be factored further using integers.

Q: Why is factorisation important?

A: Factorisation is crucial for simplifying expressions, solving equations (especially quadratic and higher-degree equations), simplifying fractions, and performing many other algebraic manipulations. It simplifies complex problems, making them easier to solve and understand.

Q: Are there different methods of factorisation?

A: Yes, various methods exist, depending on the complexity of the expression. These include the greatest common factor method (as shown above), difference of squares, grouping, and the quadratic formula (for quadratic equations). The choice of method depends on the specific expression.

Q: Can I factorise expressions with more than one variable?

A: Absolutely. The principle remains the same: identify the GCF considering both numerical coefficients and variables.

Conclusion

Factorisation is a fundamental algebraic skill with widespread applications. The process of factorising an expression involves identifying the greatest common factor among the terms and rewriting the expression as a product of the GCF and the remaining terms. This process simplifies expressions, making them easier to manipulate and solve more complex mathematical problems. Understanding the underlying principles, such as the distributive property and greatest common divisor, is key to mastering factorisation. While we focused on the simple example of 5x + 15, the principles demonstrated here are applicable to a wide range of algebraic expressions, paving the way for more advanced mathematical concepts. Through practice and understanding, factorisation can become a straightforward and indispensable tool in your mathematical toolkit.