Factoring sum of cubes

factor 27 X to the sixth plus 125 so this is a pretty interesting problem and frankly the only way to do this is if you recognize it as a special form and what I want to do is kind of show you the special form right first and then we can kind of pattern match so the special form is if I were to take and this is really just something you need to know you know that I'd argue whether you really need to know this but it actually do this problem it's something you just need to know and that's if you have a squared minus a B plus B squared and you multiply that times a plus B let's think about what we're going to get so we're going to take a product right here we're multiplying so let's do some algebraic multiplication so let's multiply B times B squared it's B to the third B times negative a B is negative a B squared B times a squared is a squared B now let's multiply this top term times a a times B squared is a b squared a times negative a b is negative a squared B and then a times a squared is a to the third and then we just have to add up all of the terms we have a negative a squared B we have a positive a squared being a negative a squared B so these guys cancel out we have a negative a B squared and a positive a B squared these guys cancel out so all we're left with is an a to the third here and a to the third and then plus plus this B to the third plus this B to the third or another way to think about it if someone gives you a to the third plus B to the third this can be factored into these two expressions that can be factored into a plus B times a squared minus a B plus B squared so this is essentially the special form if you have a sum of cubes it can be factored out as the sum of the the cube roots plus or the sum of the cube roots times this expression right here and we just showed that it works so let's see if we have that special form here well 27 is definitely the cube of 3 3 the third power is 27 X to the sixth is also the cube of x squared if you raise X to the 6 to the 1/3 power you get x squared so this first term right over here can be rewritten as 3x squared to the third power and the second term right here that's 5 to the third power so plus 5 to the third power and just this might be a little bit confusing for you so just let's never hurts to review let's multiply 3x squared times 3x squared times 3x the squared that is literally equal to 3 times 3 times 3 times x squared times x squared times x squared that's this part right here is 27 x squared times x squared is X to the fourth times x squared is X to the sixth or you could just raise both of these to the third power 3 to the third is 27 x squared to the third power you take an exponent to an exponent and you're going to take the product of the exponents so it'll be X to the 2 times 3 or X to the sixth power so now we know that we have this pattern so we can just use this we have the sum of cubes so just by using this pattern right over here that means that we can factor it as this is going to be equal to 3x squared that's our a let me make it clear this right here is our a this right here is our B so it's going to be a plus B so it's going to be 3x squared plus B plus 5 times a squared a squared let me do this in a new color so 3x squared squared let's write think about that for a second 3x squared squared well that's going to be 9x to the fourth so it's going to be times 9x to the fourth minus the product of these two things so minus the product of 5 and 3x squared so minus 15 x squared and then finally plus B squared B is 5 so it's going to be 5 squared so plus 25 when I say B is that this is B not the whole five to the third and when I say a just this part is a and we're done and we I won't explain it in detail in this video but this right here for thinking about real numbers we can't actually factor this anymore so we are done factoring this and remember this is really just a very very very special case of being able to recognize the sum the sum of cubes