Factoring Sum of Cubes

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Factoring Sum of Cubes

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Factor twenty seven x to the sixth plus one twenty five
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 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 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 gonna 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 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 -
Oh, we have a positive a squared b
and 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
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 cube roots
plus- er, 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 three
three to the third power is twenty seven
x to the sixth is also the cube of x squared
if you raise x to the sixth to the one third power
you get x squared
So this first term right over here can be rewritten
as three x squared to the third power
and the second term right here-
that's five to the third power
so plus five to the third power
And this might be a little bit confusing for you
so just let's - never hurts to review
let's multiply three x squared times
three x squared times three x squared
that is literally equal to three times three
times three times x squared times x squared
times x squared
that's this part right here is twenty seven
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
three to the third is twenty seven
x squared to the third power -
you take an exponent to an exponent
and you're gonna take the product
of the exponents
so it'll be x to the two times three 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
three x 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 three x squared plus
b
plus five times a squared
a squared
let me do this in a new color
so three x squared squared
let's start thinking about that for a second
three x squared squared
well that's going to be nine
x to the fourth
so it's going to be times nine x to the fourth
minus the product of these two things
so minus the product of five and three x squared
so minus fifteen x squared
and then finally plus b squared
b is five
so it's going to be five squared
so plus twenty five
when I said b is
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 could
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

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