WEBVTT
00:00:00.440 --> 00:00:05.520
Expand and fully simplify brackets 𝑥 plus eight brackets 𝑥 minus seven.
00:00:06.920 --> 00:00:09.800
Expanding in algebra means multiplying.
00:00:09.800 --> 00:00:15.520
We need to multiply each of the terms in the first bracket by each of the terms in the second bracket.
00:00:15.520 --> 00:00:18.160
There are two methods we can use to complete these expansions.
00:00:19.320 --> 00:00:22.440
The first method usually called FOIL or sometimes FOIL face.
00:00:22.840 --> 00:00:24.320
We’ll see why it’s called that in a moment.
00:00:25.240 --> 00:00:28.800
In the word FOIL, each letter stands for something.
00:00:28.800 --> 00:00:34.600
F stands for First, O stands for Outer, I stands for Inner, and L stands for Last.
00:00:35.360 --> 00:00:40.920
These tell us what order we can perform our multiplication in so that we don’t lose any terms.
00:00:40.920 --> 00:00:42.040
Let’s see what this looks like.
00:00:42.880 --> 00:00:46.040
The first two terms in our brackets are 𝑥 and 𝑥.
00:00:46.040 --> 00:00:48.120
So we’re going to multiply these terms together.
00:00:49.120 --> 00:00:52.240
When we multiply a number by itself, we’re squaring it.
00:00:52.240 --> 00:00:55.000
So 𝑥 multiplied by 𝑥 is 𝑥 squared.
00:00:55.920 --> 00:01:00.040
We’re now going to multiply the outer terms; that’s 𝑥 and negative seven.
00:01:01.240 --> 00:01:04.840
Be really careful here: this is a negative seven, not a positive seven.
00:01:06.240 --> 00:01:08.600
𝑥 multiplied by seven is seven 𝑥.
00:01:08.600 --> 00:01:11.800
So 𝑥 multiplied by negative seven is negative seven 𝑥.
00:01:13.280 --> 00:01:16.720
We’re next going to multiply the two inner terms; that’s eight and 𝑥.
00:01:17.720 --> 00:01:20.080
And that gives us eight 𝑥.
00:01:20.080 --> 00:01:25.680
Finally, we’re going to multiply the last terms in each bracket; that’s eight and negative seven.
00:01:25.680 --> 00:01:28.920
A common mistake here is to think that we need to add these terms.
00:01:28.920 --> 00:01:32.120
In fact, remember we said that expand means multiply.
00:01:32.120 --> 00:01:34.560
So we’re going to multiply eight by negative seven.
00:01:35.480 --> 00:01:38.800
And that gives us negative 56.
00:01:38.800 --> 00:01:41.720
There’s a clue in our question that tells us we’re not quite finished.
00:01:41.720 --> 00:01:43.880
We need to fully simplify our expression.
00:01:44.880 --> 00:01:48.640
And when we simplify, we collect like terms.
00:01:48.640 --> 00:01:51.160
Now, remember 𝑥 squared is different to 𝑥.
00:01:52.200 --> 00:01:55.760
So we have 𝑥 squared at the front of our expression.
00:01:55.760 --> 00:01:58.920
We then need to add negative seven 𝑥 and eight 𝑥.
00:01:59.880 --> 00:02:04.600
We can use a number line and just begin by imagining we’re adding negative seven and eight.
00:02:05.440 --> 00:02:07.960
To do this, we’d start at negative seven.
00:02:07.960 --> 00:02:16.960
Because we’re adding eight, we’re going to move up the number line: one, two, three, four, five, six, seven, eight.
00:02:18.280 --> 00:02:20.960
That means negative seven plus eight is one.
00:02:21.680 --> 00:02:25.000
So negative seven 𝑥 plus eight 𝑥 is one 𝑥.
00:02:26.160 --> 00:02:28.400
Remember though we don’t need to write the number one.
00:02:28.400 --> 00:02:32.880
We can simply write 𝑥 and then we have negative 56.
00:02:32.880 --> 00:02:38.200
So this expression expands and simplifies to 𝑥 squared plus 𝑥 minus 56.
00:02:39.400 --> 00:02:42.800
Did you spot why this is sometimes called FOIL face?
00:02:43.040 --> 00:02:51.600
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.
00:02:51.600 --> 00:02:52.960
Now, let’s look at the second method.
00:02:53.960 --> 00:02:57.800
This is the grid method and it works much like the grid method for long multiplication.
00:02:58.600 --> 00:03:03.280
Here, I’ve put 𝑥 plus eight at the top of the grid and 𝑥 minus seven at the side.
00:03:03.480 --> 00:03:06.680
But it actually doesn’t matter which order we choose.
00:03:06.680 --> 00:03:08.120
Let’s figure out what goes here.
00:03:09.040 --> 00:03:13.160
To get this term, we multiply 𝑥 by 𝑥; that’s 𝑥 squared.
00:03:13.840 --> 00:03:18.520
Then here, 𝑥 multiplied by eight is eight 𝑥.
00:03:18.760 --> 00:03:22.480
We then have negative seven 𝑥 here and negative 56 here.
00:03:23.320 --> 00:03:28.480
Notice how the terms in our grid are the same as the terms we achieved when we used FOIL.
00:03:28.800 --> 00:03:35.960
Once again, by adding the like terms, we end up with 𝑥 squared plus 𝑥 minus 56.
00:03:35.960 --> 00:03:38.880
Either method for expanding double brackets is absolutely fine.