one we have a quality expression in the expanded form and ideally speaking we would like to factor
this as a product of two quadratics but the main question is how do we do that right well let's
look at Eco case first namely how do we factor a quadratic let's say x squared plus 5x Plus 6.
well one way to do this is the cross method and that's just you break down x squared
which is x times x and then we look at 6 1 times 4 give us six let's use two and three
why not six times one right well we take this times that which is 3x and we take this times
that which gives us 2x together we do get a 5x so we know we have the correct combination the factor
is X plus 2 here and then we have another One X plus three so this right here is the cross method
and now the question is can we do the similar thing for this right here and
the answer is yes in fact I thought about this on my own so I was pretty proud of it
so let's see how we can actually make this work first let's do KX to a force Power the
only way is x square times x squared that's for sure thing for the this is the one right but
if it's not one yeah it's just going to be more bizarre that's all
now I would like to go to the middle term right here remember this is going to be
a factor in the quadratic form likewise this right here so this part is going to be with x
now we will just have to ask ourselves what should we put right here and right here do we look at the
12 and ask ourselves what times will give us 12 that 2 times 6 3 times 4 will mean maybe negatives
in fact no we cannot just look at this term in that term and then use 12. no right now I'm just
going to put down a and b and it's not true that to say a times B has to be equal to 12. this right
here is not true because this is not the only way to produce this term we will also have to
consider the x squared term right here times the constant term for the second factor and likewise
this constant term times this x square so these numbers are going to matter too so right now we
just put on Ax and also BX what good does this do well this is the part that we can do the cross
method so this is like the first cross method and we do another one this is the double cross method
so right here I'm just going to take this multiply by that and we'll get BX to the third power and
then we take this times that we get ax to the third power in fact these are the only two ways
to produce the X Plus third power terms so we must have this plus that being equal to negative four X
plus their power so just look at the coefficient we must have a plus b being equal to negative 4.
now let's look at the constant term which is 15 and for this right here we just have to kind of
guess and check like what we did right here and hope for the best thing I know it's like 1 times
15 no don't do that let's trust two times five so let me put down three here and also 5 here
what do we do next though where we do this take this times that which is 5 a x take this times
that which is 3 b x in fact these are the only two ways to produce the X term so we must have
this coefficient plus that coefficient equals that so we can write this down as 5 a plus 3B
equals negative 16. and now as we can see we have a system of equations and let's stop it real quick
let's multiply Everybody by negative 3 here and that will give us 12 here and then we just
add them so they cancel so this and that is 2A equals this and that is going to be negative 4.
so that means a is equal to negative 2. and when a is equal to negative 2 plugging here we know B
is also just happened to be negative 2. all right this is wonderful huh but does this really work
there's one number that we haven't used yet and that's the 12 right here so this is how we do it
we can put back the negative 2 here and also the negative 2 here all right and what we do
first is we multiply this to and you can just write down the coefficient but just to be clear
I will also write down the X term so this times that is going to be positive 4x squared and then
we'll take this multiply with this which is the 5 x square and then one more from here to here we
multiply we get 3 x squared and together when we add them up we know we have 12x squared we got it
I know it works also wonderfully right yeah because I make this yeah but anyway though
the fact here for this is just reading across so this is equal to x squared minus 2X and then
plus 3 times the other factor X squared minus 2x plus 5 and ladies and gentlemen here we have it
this is the factor for the first one now for the sake one we noticed that this is just from the
first question and then at the end here we minus the three yeah I made all these questions yeah
so one way to do this is just multiply everybody and then minus three and do the double cross
method again but we don't have to because we have this in this particular form for a reason notice
this right here is x squared minus 2x likewise this but for the first one we have plus five and
the second one we have plus three they're just off by two and to make everything much easier
for us to follow we can just do a substitution so the second method here is the substitution
when the expression is nice this data I'm going to let you to be my new variable I'll just call that
to be the first parenthesis here so x square minus 2x plus 5. so this expression becomes U times
well this right here is U but -2 because this is three right just minus two here minus two here
and then don't forget we still have that minus 3 here now this is going to be so much easier
first for us to handle distribute and order so we get U squared minus 2 U and then minus three
and how do we do this yeah the little cross method I showed you earlier but we can do
this in our head this right here is just going to be U minus 3 times U plus one
what's you use that right so on oh we have what x square minus 2x 5 minus 3 we have plus two and
then this one x square minus two x five plus one so we have plus six and you know what we are done
yeah so thanks to this particular form we can do a substitution okay this looks like a special case
huh because we have X cubed and then this is a perfect square and then just plus a number this
is still a quality expression if you really want to we can just multiply this out and then combine
items and then do the double cross again but I'm going to do that to you guys it works though
that's C how can we take advantage of this kind of form again we have X
Cube and then something Square and a number this number is not really doing us any good
one way that we like to use when we are factoring polynomials is factored by
grouping and sometimes maybe we'll have to add a number with subtract numbers things like that
right so this seven is kind of suspicious this is how we can actually make it better for us
here we have a cube ideally speaking if we can somehow also have a cube number then we can
either do the sum of two cubes with the difference of two cubes here we have a perfect square so if
we have another square number maybe you can do the difference of two squares so let's take a look at
is it possible for us to find a combination of this and that and we get seven
plus or minus Maybe yes look at this eight minus one gets a seven so let's go ahead and write the
seven as eight minus one and I'm going to put the eight next to the X Cube so that's D4 will happen
okay so here we have X cubed and then we are going to add a to it and then I'm just going
to keep this right here x square minus 2x plus 5 squared but we also need to have the minus one
now this is so much better because you'll see this right here we can just Factor it as the
sum of two cubes so we will get X plus two times x squared minus two x and then plus four done
then we are going to add for the second term
this part yeah we can just Factor this as the difference of two squares so we take
this minus one so we'll just have x squared minus 2X plus 5 minus 1 which is plus four
and again this right here is just because five minus one and the other factor from here is x
squared minus two X plus five plus one so that would be plus six so again this is five plus one
what do we see yes notice these words are so nicely
this and that and I know a lot of these factoring questions they're all like purpose the setup so
that it works out like that but hopefully I showed you guys the thought process and that's how you
approach solving like difficult questions right so how to make things happen that's hard to say it
all right so we're going to put that out so ladies and gentlemen we will have the first
factor and that will be the x square minus 2x plus 4 and then the second factor is this
combining with that so let's put down x to the second power first and then here we have the
minus 2X plus X so that would be minus X and that's the Plus 6. plus 2 that will be plus eight
so again we've added it and I should have told you guys that none of the quadratic factors that
you guys see in these videos are his factorable so that's why I didn't bother with any of that