Welcome to Georgia Highlands College Math 97 and Math 99 video tutorials.
In this video segment, we'll be answering the question, how do you factor a sum
of cubes? Well, first of all, a sum of cubes is a binomial in the form of sum
term cubed plus sum term cubed.
The first step in this process is to actually identify that it fits into this
mold of being a sum of cubes. So you should try to rewrite
your binomial as a monomial cubed plus a monomial cubed.
This second step in this process is to set up your parentheses for your factored
form. Now when we do so, we should remember to always use SOAP.
I know that sounds a little crazy, but obviously SOAP stands for something and
it stands for Same Opposite Always Positive, which still might be a little bit
confusing.
Well, SOAP is talking about the signs that you place inside your parentheses. So
since the sign that we begin with is a plus sign, the same would be another plus
sign and that goes in the first set of parentheses. The second set of
parentheses starts with an opposite sign, so subtraction, and then the last sign
is always positive. So we've set up our factored parentheses here.
Now the pattern that this follows when you factor the sum of cubes is there are
singles of the monomials A and B out front. So it would simply just be A and B.
Then the second statement here says square ends. So you'd want to square A at
one end of the second set of parentheses and square B at the other end of the
second set of parentheses.
And then our final directive in setting up our factored form is that singles are
in the center. So it's just the product of A and B in the center of the second
set of parentheses. You simplify after you put things into place and then you
end up checking your product with multiplication to see if you start out with
the same, or end up with the same binomial that you started out with.
Let's take a look at an example.
Here you can see we have the example of X cubed plus 27, and the first step in
our process is just to check and see if this actually fits
the mold of being a sum of cubes.
So, I want to see if I can write this same binomial in the form of something
cubed plus something cubed.
Well, it's pretty clear to see that for that first term X cubed, we would simply
have to cube X to make that term. And the second term 27 we would have to cube
3. So since we can fit X cube plus 27 into the mold of something cubed plus
something cubed, this satisfies the requirements for being a sum of cubes. Now
we're ready to start the actual factoring process.
So we begin by just giving ourselves parentheses to land in, for our factored
form. So now we use SOAP.
Same
Opposite
Always
Positive
All right, we're ready to place singles out front, so I have X here
and 3 here. Now we have square ends. So X squared out front and 3
squared on the end.
Please notice that I plugged these into parentheses before I squared them. You
may have a situation where you have multiple factors within that term that need
to be squared and so using parentheses when you plug into these positions helps
you see what all gets squared in the end.
And finally, we have singles in the center. So it's just the product of A and B
which in our case here is X and 3.
So we have this thing in its' factored form now. All we have to do
is simplify it and clean it up a little bit. Well nothing changes about our
first set of parentheses here. It's still X plus 3, there's no more
simplification to do.
However, a little bit can be done in the second set of parentheses. X squared
gives us X squared. Negative X times 3 gives the term negative
3X and 3 squared gives us plus 9. And this is the factored form of the binomial
which is a sum of cubes, X cubed plus 27.
But we should check this before we make that statement. So I'm just going to
check that when I multiply these factors with each other that I end up with that
same binomial that I began with and I do this with distribution. So I distribute
each term in the first polynomial to each term in the second polynomial.
All right, so X times X squared gives X cubed. X times negative 3X is negative
3X squared. X times 9 is positive 9X. Coming back I'm going to distribute my 3
now. 3 times X squared is positive 3X squared. 3 times negative 3X is negative
9X and 3 times 9 is positive 27.
When you simplify this, you come back and combine your like terms. However, when
you do so, we notice that we have two pairs of opposite terms.
A negative 3X squared and a positive 3X squared combined to make zero, and
positive 9X and negative 9X also combined to make zero.
So in the end, after all like terms are combined, we end up with X cubed plus 27
and if you notice, we can verify sure enough, we end up with the same binomial
that we started out with.
Therefore, we can safely say that X plus 3 times X squared minus 3X plus 9 is
the factored form of the sum of cubes, X cubed plus 27.
I hope that this has been helpful for you in understanding how to factor a sum
of cubes. If you have any other questions regarding this topic, please contact
your Highlands instructor.
Thank you.
