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Factoring sum of cubes

Sal factors 27x^6+125 as (3x^2+5)(9x^4-15x^2+25) using a special product form for a sum of cubes. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Factor 27x 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 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. And I'd argue whether you really need to know this. But actually to do this problem, it's something you just need to know. And that's if you have a squared minus ab 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 is b to the third. b times negative ab is negative ab squared. b times a squared is a squared b. Now let's multiply this top term times a. a times b squared is ab squared. a times negative ab 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 positive a squared b and a negative a squared b. So these guys cancel out. We have a negative ab squared and a positive ab squared. These guys cancel out. So all we're left with is an a to the third here and then 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 ab 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 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 to the third power is 27. x to the sixth is also the cube of x squared. If you raise x to the sixth 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 this might be a little bit confusing for you. It never hurts to review. Let's multiply 3x squared times 3x squared times 3x squared. That is literally equal to 3 times 3 times 3 times x squared times x squared times x squared. 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. Let me do this in a new color. So 3x squared squared-- let's 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 15x squared. And then finally plus b squared. b is 5. So it's going to be 5 squared, so plus 25. And when I said b is-- this is b, not the whole 5 to the third. And when I say a, just this part is a. And we're done. And I won't explain it in detail in this video. But this right here, if we're thinking about real numbers, we can't actually factor this any more. 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 of cubes.