in this video we're covering how to factorize an expression using just a single bracket like factorizing the expression 8x squared plus 12x into 4x brackets 2x plus 3. if you already know how to expand brackets then this is basically the exact opposite because we're going from the expanded version to the bracketed version the idea behind factorizing is you have to find the highest factor that all of the terms in your expression have in common and then take that common factor out of the expression to be on the outside of a set of brackets now once you get used to it factorizing can become quite easy and you can start to skip a few steps but in this first example we're going to take it slowly and fully explain everything so to factorize 8x squared plus 12x the first thing we need to do is find the highest common factor of the 8x squared term and the 12x term the best way to do this is to start with the numbers and then look at any letters one by one so starting with the eights and the twelve you might just know off the top of your head that the biggest number that goes into both of them is four if you didn't know that though then you'd have to list out all the factors of each of them and pick the biggest factor that's in both lists the best way to do this is to find the factor pairs of each so for eight the factor pairs are one times eight and two times four so its factors are one two four and eight then for 12 its factor pairs are 1 times 12 2 times 6 and 3 times 4 so its factors are 1 2 3 4 6 and 12. and then by comparing these two lists we can see that the biggest number that occurs in both of them is the four so that's the highest common factor of 8 and 12.
next you need to look at the letters and see if there are any common factors in those this question only has x's in and we can see that the first term has an x squared which means x times x and the second term just has a single x so 1 times x this means that the biggest number of x's that they both have in common is just a single x so the highest common factor here is just x so at this point we found that both the 8x squared term and the 12x term of our expression have a factor of four in common and a factor of x in common so putting these together we can see that the overall common factor must be four x now that we've found the common factor we need to take it and place it on the outside of a set of empty brackets inside the bracket we need to write the other factors that are needed to make our original terms so we can start by thinking what do we need to multiply 4x by to get 8x squared well to get from 4 to 8 you would have to multiply by 2 so we put 2 in our bracket and to get from x to x squared you need to multiply by x so we put an x in there as well making our factor inside the bracket 2x then we need to do the same thing for our 12x term to get from 4 to 12 we need to multiply by positive 3 and to get from x to x we don't need to do anything so the factor inside the bracket is just plus 3. this means that our overall answer is 4x brackets 2x plus 3. and to check its right you can just expand the bracket back out and see if it equals 8x squared plus 12x so once we've drawn our little arrows to show that we're going to multiply 4x by both the terms inside the bracket we can do 4x times 2x which is 8x squared and 4x times 3 which is 12x and this is exactly what we started with so we know that our answer is definitely correct now that example was fairly long-winded so let's try a couple more examples but a bit more quickly for this first one we're trying to factorize 9 a b plus 15 b squared first we look at the 9 and the 15 and think of the biggest factor that they both have in common which is a three next we need to look at the a's but there's no a term in 15 b squared so we know that there won't be an a in our common factor if we look at the b's though we can see that 9 a b contains 1 b and 15 b squared effectively has two b's being multiplied together because 15b squared means 15 times b times b so both terms have one b in common which means that we can add a b to our common factor to make it 3b next we place this 3b on the outside of some empty brackets and to figure out what's going to go inside the bracket we need to work out what 3b has to be multiplied by to make 9ab and 15b squared to make nine a b we're gonna have to times it by three a so we write three a inside our brackets and to make 15 b squared we need to multiply it by 5b so we also have to put a plus 5b in our bracket as well giving us a final answer of 3b brackets 3a plus 5b for a last example let's try this one where we've got three terms instead of two the technique we use is exactly the same so the first thing you need to do is find the highest common factor of all the numbers so of 15 10 and 20 which is five next we can move on to the x's and the highest number of x's that they all have in common is just a single x so we add that to our 5.
