Expand and fully simplify brackets 𝑥 plus eight brackets 𝑥 minus seven.
Expanding in algebra means multiplying. We need to multiply each of the terms in the first bracket by each of the terms in the second bracket. There are two methods we can use to complete these expansions. The first method usually called FOIL or sometimes FOIL face. We’ll see why it’s called that in a moment.
In the word FOIL, each letter stands for something. F stands for First, O stands for Outer, I stands for Inner, and L stands for Last. These tell us what order we can perform our multiplication in so that we don’t lose any terms. Let’s see what this looks like.
The first two terms in our brackets are 𝑥 and 𝑥. So we’re going to multiply these terms together. When we multiply a number by itself, we’re squaring it. So 𝑥 multiplied by 𝑥 is 𝑥 squared.
We’re now going to multiply the outer terms; that’s 𝑥 and negative seven. Be really careful here: this is a negative seven, not a positive seven. 𝑥 multiplied by seven is seven 𝑥. So 𝑥 multiplied by negative seven is negative seven 𝑥.
We’re next going to multiply the two inner terms; that’s eight and 𝑥. And that gives us eight 𝑥.
Finally, we’re going to multiply the last terms in each bracket; that’s eight and negative seven. A common mistake here is to think that we need to add these terms. In fact, remember we said that expand means multiply. So we’re going to multiply eight by negative seven. And that gives us negative 56.
There’s a clue in our question that tells us we’re not quite finished. We need to fully simplify our expression. And when we simplify, we collect like terms. Now, remember 𝑥 squared is different to 𝑥. So we have 𝑥 squared at the front of our expression. We then need to add negative seven 𝑥 and eight 𝑥. We can use a number line and just begin by imagining we’re adding negative seven and eight.
To do this, we’d start at negative seven. Because we’re adding eight, we’re going to move up the number line: one, two, three, four, five, six, seven, eight. That means negative seven plus eight is one. So negative seven 𝑥 plus eight 𝑥 is one 𝑥. Remember though we don’t need to write the number one. We can simply write 𝑥 and then we have negative 56. So this expression expands and simplifies to 𝑥 squared plus 𝑥 minus 56.
Did you spot why this is sometimes called FOIL face? When we draw these arrows in to remind us of the order in which to multiply, they look a little bit like a face: we have two eyebrows, a nose, and a mouth.
Now, let’s look at the second method. This is the grid method and it works much like the grid method for long multiplication. Here, I’ve put 𝑥 plus eight at the top of the grid and 𝑥 minus seven at the side. But it actually doesn’t matter which order we choose. Let’s figure out what goes here.
To get this term, we multiply 𝑥 by 𝑥; that’s 𝑥 squared. Then here, 𝑥 multiplied by eight is eight 𝑥. We then have negative seven 𝑥 here and negative 56 here. Notice how the terms in our grid are the same as the terms we achieved when we used FOIL. Once again, by adding the like terms, we end up with 𝑥 squared plus 𝑥 minus 56.
Either method for expanding double brackets is absolutely fine.