Factorising algebraic expressions might seem daunting at first, but mastering this skill is crucial for success in higher-level mathematics. This comprehensive guide breaks down the process of factorising expressions like 7 + 7xy + xyz, providing you with the building blocks you need to confidently tackle similar problems.
Understanding Factorisation
Before diving into the specifics of factorising 7 + 7xy + xyz, let's establish a solid understanding of the core concept. Factorisation is essentially the reverse of expanding brackets. It involves expressing a given algebraic expression as a product of simpler expressions (its factors). For example, expanding (x+2)(x+3) gives x² + 5x + 6; factorising x² + 5x + 6 gives (x+2)(x+3).
Identifying Common Factors
The first step in factorising any expression is to look for common factors among the terms. This involves identifying factors that are present in each term of the expression. This is often the easiest way to simplify the expression and make further factorisation easier.
Factorising 7 + 7xy + xyz: A Step-by-Step Approach
Let's now tackle the specific expression: 7 + 7xy + xyz. Notice that 7 is a common factor in the first two terms (7 and 7xy). We can begin the factorisation process by pulling out this common factor:
7(1 + xy) + xyz
Unfortunately, at this point, there isn't an obvious common factor across all terms. This expression is already in its simplest factorised form using common factors. This expression does not factor further using simple techniques. More advanced techniques would be necessary to fully factor this, and would involve considering more complex factorization strategies which are typically covered at a higher mathematics level.
Expanding Your Factorisation Skills
While 7 + 7xy + xyz doesn't offer a straightforward complete factorisation using elementary techniques, understanding the process of finding common factors is a critical first step. Practice identifying common factors in various expressions. Once comfortable, you can explore more advanced factorisation techniques such as:
- Grouping: This technique involves grouping terms with common factors to enable further factorisation.
- Quadratic Factorisation: Learning to factorise quadratic expressions (expressions of the form ax² + bx + c) is essential.
- Difference of Squares: Understanding and applying the difference of squares formula (a² - b² = (a+b)(a-b)) is another valuable tool.
- Cubic and Higher-Order Polynomials: Factorising cubic and higher-order polynomials often requires more advanced algebraic manipulation and potentially the use of polynomial long division.
Mastering Factorisation: The Path to Success
Mastering factorisation isn't just about memorising formulas; it's about understanding the underlying principles and developing a systematic approach. Start with simpler expressions and gradually work your way up to more complex ones. Practice regularly, and don't hesitate to seek help when needed. With consistent effort and a solid understanding of the basics, you'll be well on your way to confidently factorising a wide range of algebraic expressions. Remember, perseverance is key!
