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# Factoring sum of cubes

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.