Look at an example of (a + b) ^ 2. Bit 2 means “square” or multiplied by itself. So basically basically a fantastic way to tell us to take two batches (a + b) and propagate them together:
(a + b) (a + b)
Expanding brackets mean multiplication. It is important to keep this in mind, as in the above expression, there are no signs between the brackets. Therefore, we actually mean
(a + b) x (a + b)
Similarly, when we multiply a by b, we simply write ab to mean ax b. The reason why we don’t show the multiplication sign in algebra is because mathematicians are lazy, and since multiplication is used so much in algebra, they cannot be bothered to write “x” all the time and can also be easily confused with algebraic the letter x, which is very often used in algebra.
So now it is time to expand these brackets. When you multiply parentheses, you need to multiply each term in the first bracket with each term in the second bracket. This is shown below:
(a + b) (a + b) = axa + ab + ba + bxb
Now we can remove this a bit. Outside we have axa and bx b. We can write them as a ^ 2 and b ^ 2. This is the opposite of what we did at the very beginning. Also, ab = ba (you can try it with numbers — for example, 7 x 8 = 8 x 7). So, we have two ab lots, which we can rewrite as 2ab. Our expansion becomes
(a + b) (a + b) = a ^ 2 + 2ab + b ^ 2
Excellent! We have successfully expanded (a + b) ^ 2! The first part of the expansion that got us axa + ab + ba + bxb may have been a bit complicated, or difficult to remember. There are various ways of remembering how to do this first part of the extension (not all of them are good), but there are several civilized ways to do this, shown below:
Method 1: FOOL
This is an interesting little mnemonic that will help you learn how to expand parentheses. It happens like this:
F means First - multiply the first term (letter) in each bracket together.
O means Outside - multiply the outer two members together.
I speak for Inside - several times inside two terms.
L means Last - multiply the last two terms in each bracket together.
Method 2: Multiplication Tables
If you thought that you got rid of them when you finally know all your tables, then you are wrong! Even if you do math at the university level, you are in the process of studying group theory. No, you haven't finished with them yet. Times tables for life, not just for primary school. Here we go ...
X, , b
a ^ 2 ab
b ba b ^ 2
Hope this didn't bring too many painful memories. At least now, armed with an understanding of how to write algebraic expressions and expand parentheses in various ways, you have on hand the most fundamental skills that give you expand Your mathematical knowledge in various fields!
