Ex 3: Factor a Sum or Difference of Cubes

INSTRUCTOR: Welcome to one more example of factoring a sum or difference of cubes. We've already reviewed these formulas. So let's take a look at our example. 

We have 64x to the sixth plus 216y the third. And this is about as challenging as they get. The reason I say this is 64x to the sixth is a perfect cube, and so is 216y to the third. 

We can write 64x to the sixth as 4x to the second raised of the third. And we can also write 216y to the third as 6y raised to the third power. However, this would be a mistake. 

Notice that 4 and 6 do share a common factor of 2. So we skipped the first step in factoring, which is to factor out the greatest common factor of our binomial. And it's important that we don't forget this first step, because if we factored this binomial, using the sum of cubes formula, with the binomial written in this form, it would not be factored correctly. So we're going to start by factoring out the greatest common factor of 64 and 216 from our binomial, which is not an easy question. 

So to do this, we're going to find the prime factorization of 64 and 216. So for 64, we can break this down into 8 times 8. And then 8 is equal to 4 times 2, where 2 is prime. And, of course, is 2 times 2. 

So the prime factorization of 64 is going to be 6 factors of 2. And now for 216, we'll start with 2 times 108. 2 is prime. 

108 is 2 times 54. 2 is prime. 54 would be 2 times 27. 2 is prime. 

27 would be 3 times 9. 3 is prime. And 9 is equal to 3 times 3. Both of these are prime. So for 216, we have three factors of 2 and three factors of 3. 

And now we can see the common factors are going to be three factors of 2. So we'll start by factoring the GCF of 8 out of our binomial. So if we do this, we're going to have 8 times-- if we factor out 8 from 64, we can see here, we're going to be left with 8 and then x to the sixth plus-- if you factor out 8 from 216, we can see we're left with 3 times 3 times 3 or 27y to the third. 

Now from here, looking at our binomial factor, we should recognize that 8x to the sixth is a perfect cube. And so is 27y to the third. The reason 8x to the sixth is a perfect cube, because if we have 2x to the second raised to the third, this will equal 8x to the sixth. And 27y to the third is a perfect cube, because if we have 3y raised to the third power, this is equal to 27y to the third. 

Looking at our binomial written in this form, referencing our formula, this helps us to recognize that a is equal to 2x squared. And b is equal to 3y. So now our sum of cubes will factor into a binomial random trinomial. 

So we'll still have this extra factor of 8. And we'll have a binomial as a trinomial. Our binomial factors going to be a plus b, which in this case would be 2x squared plus 3y. 

The first term in the trinomial factor is a squared. So if a is equal the 2x squared, a squared would be equal to 2x to the second squared, which would be 4x to the fourth minus a times b-- well, a, again, is equal to 2x squared. And b is equal to 3y. 

So 8 times b would be 6x squared y. And then we have plus b squared. So if b is equal to 3y, b squared would be equal to 3y to the second, which would be 9y to the second. 

And now our expression is factored completely. Remember, when applying the sum or difference of cubes formula, this trinomial will not be factorable. OK. I hope you found this helpful.