# How to expand brackets algebraically

by | Apr 5, 2023

In this blog post we are going to show you how to expand brackets algebraically, it is a very important skill in NAT5 mathematics and is something you will need to be comfortable with in order to succeed in your exams. Expanding a bracket simply means multiplying out every term in the brackets. Some important prerequisite skills for expanding brackets will be understanding how to apply index laws. If you aren’t familiar with index laws or need a refresher, feel free to have a look at our available resources! Let’s go through a few examples together using the FOIL method, which stands for “First, Outer, Inner, Last”.

## What is the FOIL method for expanding brackets?

The FOIL method is a popular technique used to expand two binomials algebraically. It’s a mnemonic that stands for “First, Outer, Inner, Last”, which describes the four terms that need to be multiplied together in order to expand the expression. This method is commonly taught in middle and high school math classes as an introductory technique for expanding brackets. When using the FOIL method, you start by multiplying the first term of each binomial together, then the outer terms, the inner terms, and finally the last terms. You then add up the resulting four products to obtain the fully expanded expression. This process can help simplify expressions and make them easier to work with when solving equations or simplifying expressions. While the FOIL method is a quick and easy way to expand simple binomials, it may not be the most efficient method when dealing with more complex expressions or polynomials of higher degrees. In those cases, other techniques such as the distributive property or the binomial theorem may be more effective. It’s important to note that the FOIL method is just one tool in a mathematician’s toolkit, and it’s not always the best option. Being able to recognize when to use FOIL and when to use other methods is an important skill in algebra, and can help students develop a deeper understanding of the subject.

## Example one expanding brackets FOIL

As a first example, let’s consider how we might want to expand (𝑥 + 2)(𝑥 + 3) using FOIL. First – We multiply the first term in each bracket, so this will be 𝑥 × 𝑥 = 𝑥² Outer – Now we want to multiply the outer terms in each bracket, these are the numbers the furthest away from each other. In this case 𝑥 × 3 = 3𝑥. Inner – We now want to multiply the inner terms. These are the two numbers which are next to each other inside the brackets, giving us 2 × 𝑥 = 2𝑥 Last – Finally, we want to multiply the last remaining terms. 2 × 3 = 6. Now that we have multiplied all the terms, we need to add them all together. (𝑥 + 2)(𝑥 + 3) = 𝑥² + 3𝑥 + 2𝑥 + 6. All that’s left to do now is simplify, so we add “like” terms together to finally find: (𝑥 + 2)(𝑥 + 3) = 𝑥 2 + 5𝑥 + 6 Let’s try another example together. If we have a scenario like 2(𝑥 + 3), then we need to multiply the term outside the bracket by each term inside the bracket.

## Example two

First, we want to multiply the outside term by the first time in the bracket. 2 × 𝑥 = 2𝑥 Next, we want to multiply the term outside the bracket by the second term inside the bracket. 2 × 3 = 6 Just as before, we now combine these two terms. In this case, we get 2𝑥 + 6

## Example Three

Now for a final example, let’s expand (8𝑥 + 12)(3𝑥 + 2) using the foil method again. First – Multiply the first terms in each bracket 8𝑥 × 3𝑥 = 24𝑥² Outer – Now both the outer numbers in the brackets 8𝑥 × 2 = 16𝑥 Inner – Next, both inner terms 12 × 3𝑥 = 36𝑥 Last – Finally, we want to multiply the last remaining terms. 12 × 2 = 24.

## Expanding brackets and the National 5 Exams

In the mathematics National 5 course, one of the topics that students learn is expanding brackets. If you are looking for more information about the national 5 exams you can read more here.  Expanding brackets involves multiplying out algebraic expressions that are in the form of brackets. In National 5, students typically learn how to expand brackets that involve one or two terms, such as (x + 2) or (2x – 3y). Students will learn different techniques to expand these brackets, including the distributive property, the FOIL method, and the use of algebraic identities. The distributive property is a fundamental concept in algebra that states that the product of a number or variable and a sum is equal to the sum of the products of the number or variable with each term in the sum. For example, 3(x + 2) can be expanded as 3x + 6 by multiplying the 3 by both terms inside the bracket. The FOIL method is a popular mnemonic device used to expand the product of two binomials, where each binomial consists of two terms. FOIL stands for “First, Outer, Inner, Last”, and it involves multiplying the First terms of each binomial, then the Outer terms, then the Inner terms, and finally the Last terms. For example, to expand (x + 2)(x + 3) using FOIL, students would first multiply the First terms (x * x), then the Outer terms (x * 3), then the Inner terms (2 * x), and finally the Last terms (2 * 3). In addition to these techniques, students in National 5 may also learn about algebraic identities such as the difference of squares and the perfect square trinomial. These identities provide shortcuts for expanding brackets that have a certain form, and they can help simplify expressions and make them easier to work with. Overall, expanding brackets is an important skill that students learn in National 5 mathematics, as it is a fundamental concept that underpins many other areas of algebra and calculus. By mastering the different techniques for expanding brackets, students will be better equipped to solve equations, simplify expressions, and tackle more advanced mathematical problems.

## Other related maths problems

How to solve Straight Line Equations and Simultaneous Equations – what they are and how to use them. In conclusion, expanding brackets is a very important skill in NAT5 mathematics and one that you will likely need to use in your other subjects too. By following our examples in this article as well as the extra problems in the video below, we hope you can further your understanding of this essential skill. If you want more information about Central Tutors get in touch today.

#### Need help studying?

We have friendly tutors to give you support with your exams in biology, chemistry, maths and physics.