Ex 1: Factor a Sum or Difference of Cubes

NARRATOR: Now we'll look at some examples of factoring a sum and difference of cubes. Notice both a sum and difference of cubes factors into a binomial and a trinomial. So if we have a binomial in the form of a cubed plus b cubed, notice how the first binomial factor is a plus b, and the trinomial factors a squared minus ab plus b squared. And if we have a binomial in the form a cubed minus b cubed, the first binomial factor is a minus b, and the trinomial factor is a squared plus ab plus b squared. 

So if we want to factor x cubed plus 64, we need to recognize that we have a sum of perfect cubes. It might be helpful to write this in the form of x raised to the third power plus 64 is a perfect cube, because it's equal to 4 raised to the third power. In this form, we can recognize that a is going to be equal to x, and b is going to be equal to 4. And now we can apply our factoring formula. Again, this will factor into a binomial times a trinomial. 

The binomial factor will be a plus b. So in this case, we'll have x plus 4. The first term of the trinomial factor will be a squared. So if a is equal to x, a squared would be x squared minus a times b. Well, if a is equal to x and B is equal to 4, a times b would be x times 4, or 4x, plus b squared. Well, if b is equal to 4, b squared is equal to 16. If we need to, we can multiply this out to verify that it is equal to x cubed plus 64. 

Looking at the second example, we have 8x cubed minus 27. So the first thing to recognize here is that we have a difference of perfect cubes. 8x cubed is a perfect cube, because we can write this as 2x to the third. And 27 is a perfect cube, because we can write this as 3 to the third. So in this form, we know that a must equal 2x, and b must equal 3. 

And now we can factor our difference of cubes into a binomial and a trinomial. The binomial factor when we have a difference of cubes is a minus b. So here we'll have 2x minus 3. 

The first term of the trinomial factor will be a squared. Well, if a is equal to 2x, 2x squared would be 4x squared plus a times b. a times b would be 2x times 3, or 6x, plus b squared. If b is equal to 3, b square would be 3 squared, or 9. OK, hope you found this helpful.