Learn how to the expression fully factorise 12r+16 with this detailed, step-by-step guide. Ideal for students and anyone looking to improve their math skills, our guide simplifies factorisation with easy-to-follow instructions and examples.
Introduction
Factorisation is a foundational concept in algebra, allowing us to simplify mathematical expressions and solve equations more efficiently. If you’ve encountered the expression “12r + 16” in your math homework or a test and are wondering how to factorise it, you’ve come to the right place. In this article, we’ll break down the steps to fully factorise “12r + 16,” providing easy-to-follow explanations and helpful tips along the way. By the end, you’ll understand how to approach factorisation confidently and accurately.
Factorising expressions like “12r + 16” can often feel intimidating, but it’s just a matter of following a systematic approach. Let’s dive into the steps and gain a clear understanding of how to factorise “12r + 16” in a straightforward way.
What Does It Mean to Fully Factorise?
Before we start working with “12r + 16,” it’s important to understand what factorisation is. To factorise an expression means to rewrite it as a product of its factors. In simpler terms, we are looking for values or expressions that multiply together to give us the original expression.
When we fully factorise an expression, we try to express it in the simplest form possible, where no further factorisation is possible. This is especially useful in algebra, as it allows us to simplify expressions, solve equations, and even identify important properties about functions.
Understanding the Expression “12r + 16”
The expression we are working with, “12r + 16,” is known as a linear expression because it contains a variable (in this case, ‘r’) raised to the power of one. Factorising linear expressions is usually straightforward and is often a first step in mastering more complex algebraic operations.
Step 1: Identify the Greatest Common Factor (GCF)
The first step in fully factorising “12r + 16” is to identify the greatest common factor (GCF) of both terms. The GCF is the largest value that can divide both terms of the expression without leaving a remainder.
- The terms in our expression:
In “12r + 16,” we have two terms:- 12r
- 16
- Finding the GCF of 12 and 16:
To find the GCF, let’s look at the factors of each number:- The factors of 12 are: 1, 2, 3, 4, 6, and 12.
- The factors of 16 are: 1, 2, 4, 8, and 16.
Step 2: Factor Out the GCF
Now that we know the GCF of 12 and 16 is 4, we can factor out 4 from each term in the expression. Factoring out the GCF involves dividing each term by the GCF and then rewriting the expression in terms of this common factor.
- Divide each term by the GCF (4):
- For 12r:
12r÷4=3r12r \div 4 = 3r12r÷4=3r - For 16:
16÷4=416 \div 4 = 416÷4=4
- For 12r:
- Rewrite the expression:
When we factor out 4 from “12r + 16,” we get:
12r+16=4(3r+4)12r + 16 = 4(3r + 4)12r+16=4(3r+4)
Step 3: Verify Your Answer
After factorising, it’s always a good practice to double-check your work by expanding the factored expression to see if it matches the original expression.
- Expand 4(3r + 4):
Use the distributive property to expand:
4⋅3r+4⋅4=12r+164 \cdot 3r + 4 \cdot 4 = 12r + 164⋅3r+4⋅4=12r+16
Since the expanded form matches the original expression, we can be confident that our factorisation is correct.
Why Is Factorising Important?
Factorisation is not just a mathematical exercise; it has real-world applications in various fields, including engineering, physics, and finance. In algebra, factorisation allows us to simplify expressions, making it easier to solve equations or perform calculations. Mastering factorisation also helps with higher-level mathematics, including calculus and differential equations.
Fully factorising expressions like “12r + 16” provides a foundation for understanding polynomials, quadratic equations, and more complex functions. By learning how to identify and factor out the GCF, you build critical skills for tackling a wide range of algebraic problems.
Common Mistakes to Avoid When Factorising
While factorising linear expressions is straightforward, there are common mistakes that can lead to incorrect answers. Here are a few pitfalls to watch out for:
- Forgetting to Identify the GCF Properly:
It’s essential to check all possible factors to find the greatest one. Missing the correct GCF can lead to an incomplete factorisation. - Not Double-Checking Work:
Expanding the factored expression helps confirm your answer. Skipping this step might mean you overlook simple errors. - Dividing Incorrectly:
Double-check that each term is divided accurately by the GCF. Miscalculating here can lead to an incorrect final answer. - Leaving the Expression in Partially Factorised Form:
Remember, fully factorising means finding the simplest form. Ensure no further factorisation is possible.
Examples of Factorising Similar Expressions
Practicing with similar expressions can improve your confidence and skill in factorising. Here are a few examples:
- Factorise 6x + 9
- GCF of 6 and 9 is 3
- Result: 6x+9=3(2x+3)6x + 9 = 3(2x + 3)6x+9=3(2x+3)
- Factorise 8y + 12
- GCF of 8 and 12 is 4
- Result: 8y+12=4(2y+3)8y + 12 = 4(2y + 3)8y+12=4(2y+3)
- Factorise 15a + 25
- GCF of 15 and 25 is 5
- Result: 15a+25=5(3a+5)15a + 25 = 5(3a + 5)15a+25=5(3a+5)
Practicing these steps with other expressions will help you become proficient in factorisation.
Conclusion
Factorising expressions like “12r + 16” is a valuable skill in algebra, simplifying equations and providing insights into mathematical relationships. By identifying the greatest common factor, factoring it out, and verifying the result, you can confidently fully factorise any similar expression. Remember, understanding these basic steps lays the groundwork for more advanced math concepts.
Whether you’re a student working through homework or an adult revisiting algebra, mastering the process of factorisation will be an asset in solving a wide range of mathematical problems. Keep practicing with different expressions, and soon, factorising will become second nature. Happy learning!
