Fully Factorise 8q+20

Fully Factorising 8q + 20: A Comprehensive Guide

Fully factorising algebraic expressions is a fundamental skill in algebra. Understanding this process is crucial for simplifying expressions, solving equations, and tackling more advanced mathematical concepts. This article provides a comprehensive guide to fully factorising the expression 8q + 20, explaining the steps involved, the underlying mathematical principles, and addressing common questions. We'll explore different factorization techniques and delve into the reasons behind each step, ensuring a thorough understanding of the process.

Introduction: Understanding Factorisation

Factorisation, in its simplest form, is the process of breaking down a mathematical expression into smaller parts (factors) that, when multiplied together, give the original expression. It's like reverse multiplication. For example, the number 12 can be factored into 2 x 6, 3 x 4, or 2 x 2 x 3. Similarly, algebraic expressions like 8q + 20 can be factored into simpler expressions. This process is essential for simplifying complex expressions and solving equations. The goal of fully factorising an expression is to find the simplest factors that cannot be further broken down. In this case, our aim is to fully factorise 8q + 20.

Step-by-Step Factorisation of 8q + 20

The key to factorising 8q + 20 lies in identifying the greatest common factor (GCF) of the two terms. The GCF is the largest number or expression that divides evenly into both terms without leaving a remainder.

1. Identify the Coefficients and Variables: Our expression is 8q + 20. The coefficients are 8 and 20, and the variable is q.

2. Find the Greatest Common Factor (GCF) of the Coefficients: Let's find the GCF of 8 and 20. We can list the factors of each number:

   • Factors of 8: 1, 2, 4, 8
   • Factors of 20: 1, 2, 4, 5, 10, 20

   The largest number that appears in both lists is 4. Therefore, the GCF of 8 and 20 is 4.

3. Factor out the GCF: Now, we factor out the GCF (4) from both terms of the expression:

   8q + 20 = 4(2q) + 4(5)

4. Rewrite the Expression: Notice that 4 is a common factor in both terms. We can rewrite the expression by factoring out the 4:

   8q + 20 = 4(2q + 5)

This is the fully factorised form of 8q + 20. There are no more common factors between 2q and 5, so we cannot simplify further.

Mathematical Justification: Distributive Property

The process of factorisation relies heavily on the distributive property of multiplication. The distributive property states that a(b + c) = ab + ac. In our case, we applied the distributive property in reverse. We started with 4(2q + 5) and, by distributing the 4, arrived at 8q + 20. Factorisation is essentially the reverse application of the distributive property.

Exploring Different Factorisation Techniques (for more complex scenarios)

While the 8q + 20 example was straightforward, let's briefly explore some other techniques used for more complex factorisation problems. These are not directly applicable to 8q + 20 but are important for a broader understanding of factorisation:

• Difference of Squares: This technique applies to expressions in the form a² - b², which factorises to (a + b)(a - b). For instance, x² - 9 factorises to (x + 3)(x - 3).

• Quadratic Trinomials: Expressions in the form ax² + bx + c can often be factorised into two binomial expressions. This often involves finding two numbers that add up to 'b' and multiply to 'ac'.

• Grouping: This method is useful for expressions with four or more terms. You group terms with common factors and then factor out the common factor from each group.

• Using the Factor Theorem: This more advanced technique helps in factorising polynomials of higher degrees. It involves finding the roots of the polynomial and using them to determine the factors.

These techniques become crucial when dealing with more complex algebraic expressions. However, for the simple case of 8q + 20, finding the greatest common factor and applying the distributive property in reverse is sufficient.

Frequently Asked Questions (FAQ)

Q: Why is finding the greatest common factor (GCF) important in factorisation?

A: Finding the GCF ensures that we fully factorise the expression. If we don't find the largest common factor, we may leave some common factors remaining in the expression, resulting in an incomplete factorisation.

Q: Can 8q + 20 be factorised in any other way?

A: While 4(2q + 5) is the fully factorised form, you could technically factor it in other less efficient ways, like 2(4q + 10). However, this is not fully factorised because there is still a common factor of 2 within the parenthesis. The fully factorised form always uses the greatest common factor.

Q: What if the expression had more terms? How would I factorise it?

A: For expressions with more terms, you would look for the GCF of all the terms. If there is no common factor among all terms, you might need to explore other techniques like grouping, as mentioned earlier.

Q: What are some common mistakes to avoid when factorising?

A: Common mistakes include:

• Not finding the greatest common factor: This leads to an incomplete factorisation.
• Incorrectly applying the distributive property: Double-check your calculations to ensure you've distributed correctly.
• Forgetting to check your answer: Multiply the factors back together to verify that they give you the original expression.

Conclusion: Mastering Factorisation

Fully factorising algebraic expressions is a fundamental skill in algebra. The process of factorising 8q + 20, as demonstrated, involves identifying the greatest common factor (GCF) of the coefficients and then factoring it out. This procedure relies on the distributive property of multiplication. While this specific example is relatively simple, understanding the underlying principles of factorisation—and the various techniques available for more complex scenarios—is crucial for success in algebra and beyond. Remember to always check your work to ensure you've fully factorised the expression and haven't made any calculation errors. Practice is key to mastering this important skill!