Factorization by using Sum & Difference of Cubes - Polynomials Video Lecture - Class 9

hello spirits so we'll continue chapter
polynomials and in paul thomas we were
studying about the factorization so now
today we'll see one more method of doing
the pet relation that is factorization
as the sum and
difference of two cubes so this is the
another method of doing the Federation
of the sum and difference of the cubes
as you studied the identity of cubes
that is a plus B whole cube and a minus
B whole cube right apart from this a
cube plus B cube and a cube minus B cube
say cube plus b QP is a plus b into a
square minus a B plus B Square and a
cube minus B cube is equal to a minus B
into a square plus a B plus B Square so
today we'll see the earth out of the
Federation by using these identities so
we'll see first example that is a cube
plus 64 now you can see at this example
is actually this question is actually in
the form of a cube plus B cube here it
is a cube + + + 64 means 4tube so it
actually in the form of a cube plus B
cube and the effort can be written as a
plus B that is a plus 4 into a square
minus a B that is minus 4a and plus b
square that is foursquare it means 16 so
this is the normal simple method of
doing the Federation by using the
identity and as you can see in here
directly in the question that this
question directly in the form of a cube
so it is easier to do the Federation
here now we'll see the next example the
next example given s54 eq + 250 B cube
now if you see in this example the this
54 e QV it not actually in the form of
perfect cube go to 54 is what the tube
of any natural number and same way to
give please not the cube of any natural
number but b cubed a cube are the
perfect school so we cannot do the car
treasure
we know that whether until we cannot
cannot convert it into the form of
perfect you so what we need to do is if
you take two common from both these
terms it will become 27 a cube plus 125
v cube and this will be equal to now if
you see here 27 a cube has 125 BQ this
27 a cube now is actually in the form of
perpetuate that is this 27 a cube is
actually the cube of three a whole cube
in the same way plus 125 is actually the
cube of pipe so it will become 5b whole
cube and now we have converted into the
intruder form of a cube plus B cube now
we can use the identity that is 2 into
eq + V Cube is a plus B that is 3a plus
5b into a square that is three a whole
square it will become 9 a square minus a
B minus ay B means minus 3a in 25 v that
is 15 a B and plus B squared at is 5 v
square so it will become fortified v
square so in this way we need to do the
editor use identity and before using the
identity we actually have to convert it
into that form so accordingly then only
we can use identity and do the
factorization I will see one more
example is be raised to the power 6
minus 5 12 into q raise to the power 6
now this actually now if you can see
here this again is actually in the form
of a cube minus make you rock now it is
actually in the power a Q minus B cube
but we have to write it it actually in
the form of perfect cube so we can write
it as P Square whole cube minus PI over
2 cube six so it will become 8 q square
whole cube so because Phi 1 to utley the
cube of eight so NVR written q6 SQ
square and out then it will become 8 q
square whole cube so it is actually five
or two q 6 so now we have converted
the form of a Q minus B cube and now we
can use the identity is a minus B that
is p square minus 8 q square e minus B
into a square get this p square whole
square garden become p 4 plus a B that
is eight p square q square then plus B
Square where it is a 2 square by a
square 64 q4 so now this we have
converted directly into the Baba now we
have done the preparation of this but
everton it is in the form of a Q minus B
cube so in this range we need to do the
Federation of them two terms which are
actually the form of a cube plus B cube
or a cube minus b q tal LC was some more
example eh the power 7 plus e in to be
raised to the power 6 now if you can see
here this is really not in the form ax
or in the form of a cube plus DQ for a Q
minus B cube so being need to do first
we need to convert it into the dead form
and then we can do the pressurization so
if you see here a raised to the power 7
plus a B into a be raised to the power 6
so if you take a comment from here it
will become a 6 plus B 6 so now if you
can see here it is plus B 6 so now and
now be conducted actually in the form of
a cube plus B cube that will become a
square whole cube plus B Square whole
cube so this will be equal to a into a
tube now it actually in the form of a
cube plus B cube so we can inside entity
that is it must be that is a square plus
B Square into a a square say square
means a square whole square so it will
become 84 minus a B that is a square B
Square and plus B Square miss whole
square of 2 whole square of beasts so V
squared certain become be four so this
completes our characterization of this
term
so in this sense we need to understand
like in which form we need to convert
and accordingly so that we can do the
characterization so we see one more
example based on this that is 27 AQ + 1
x 64 b q + 27 a square upon poor be yes
9a upon 16 b square so accordingly we
introduce the contraction of this so
what we can do this did you see the
first object is 27 a cube is actually in
the form of a is actually the
perspective of 38 and if you see the
second term is 1 x 64 BQ is actually the
perfect you walk 1 by 4 B so now what we
can we can do is as first who comes out
in the form part redtube we need to
understand about the last two terms so
if you see the last two done so we can
write this last two terms in activating
here in the form of perfect you to make
it to use the identity here so that we
can do the factorization so if you can
write 2782 best three a whole cube plus
54 s4b whole cube plus and now in among
these two terms if you take the common
term comment among these fruit exist 9a
upon poor be so the remaining part is 3a
plus 1 by 4 B naught you can see here is
actually a cube plus B cube plus 3a be
into a plus B this actually is the
identity of a cube plus B cube for sorry
exactly I did you a plus B both you for
a plus B won't you miss it you predict
you plus 3a be into a plus B if you
remember that is a plus B volk you it's
actually a cube plus V cubed plus 3 AV
into a plus B and this and this question
we are converted actor
in the form of this AQ + b q + behavior
to a plus because if you consider this 3
as the a so it will become a cube plus 1
backbone will become B cube plus 3a be
into a plus B plus 3 abbs we protect
recover from this nine it will become
3ei upon 4v so it is actually in the
form of 3a be so that's why we can write
this whole thomas 3a plus 1 by 4 be
pulled cube and if you open it we get
the same result as the question so
accordingly sometimes we need to use
these identities also and most of all we
need to understand which identity to be
used BF so accordingly we can convert it
into that proper form and then we will
do the factorization now we will see one
more example based on this
so first I'd jump right on the cushion
shrek is a cube plus 3 a square B plus
3a v square plus B cube minus 8 so now
for bring the Federation of this term if
you see a few properly we have
visualized this cushion this first four
terms will actually in the form of a
cube plus B cube plus 3a be into a plus
B minus eight so if you see here that is
Incubus beekeeper 3b to him it must be
settle in the form of a plus B whole
cube minus eight that is too cute so now
this this so we have reverted reptile in
the form of a cube minus B cube so it
will become a minus B that is a plus B
minus 2a minus B into a square say
square means a plus B holy Sh words that
could become a square plus B squared
plus 2a be a square plus b square b
square means to square so it will become
4 and plus a piece o plus a B means plus
2a plus 2b so this is so in this way we
have done the Federation so here we have
used to identity one identity we are
used as it does me won't you the other
one we are used as a Q minus B so
sometimes we need to do this they've
also read we can use the multiple
identities in a single portion to do the
factorization so this is the another
method of doing the Federation and the
same way we'll see one more example
based on this so that is a cube minus B
cube plus 1 plus 3a be so if you observe
this is actually if you see you cannot
use finally hi Darius 8 you buy this
video because here if you use identity
if you buy this beat you by looking
after the first two terms yeah it will
not solve our purpose so for this what
we need to do to err is if you observe
here we have started
identity that is that identity was a
cube plus B cube plus C cube minus three
ABC is equal to a plus B plus C into a
square plus B Square the C square minus
a B minus b c minus CA so we can use
this identity here as if you see here
first term is a cube second term is
minus b q so if we can take it as plus
here if we can write it as minus b1 cube
the third term is 1 cube that is C cube
plus minus 3 ABC so if we can use the
minus here minus 3 into a and instead of
B you can u minus B because our second
term is minus B dot plus B and into one
that is C is 1 so it is actually in the
power of a cube plus b q + c q minus 3
NB C so accordingly we can use air is a
plus B plus C that is a minus b plus 1
that is a plus B plus C into a square
plus B Square + C square that is 1 minus
a be so minus a B means minus a into
minus B so it will become plus ad minus
BC so why not bc means b is a DS our
vital here b is minus B so minus B into
one if it will become minus B and minus
already there so it will become plus B
so a squared plus B squared plus 1
instead of minus eighty it will become
plus AP instead of minus BC it will
become plus B and minus CA minus CA
means minus a into one so it will become
minus e so in this way we have done the
factorization here this is the method of
doing the Federation so we can use any
additives but it is very important for
destin which added you to be you spare
so accordingly we can solve the question
properly in the same way we will use
we'll see one more position based on
today's practice x minus y whole cube
plus y minus Z whole cube plus
z minus X 4 cube so in this question if
you see here actually already in the
form of they are gamer a minus B volt
you if you use the IDT you might as
behold you this question is ready to
this solution of this question will
become very large and it will become
very complicated so instead of doing
this V whatever entity we have youth
will actually visualize own identity
reporting agency which I need you to be
used here so no so the identity which we
discussed the last for sure was a cube
plus DQ plus C cube minus VA bc is equal
to a plus B plus C into a square plus b
square plus z square by this eb by this
VC by just see this is the identity we
have discussion discussed in the large
last question in this question we use
the special case of this debt identity
and according to the special case of
that identity what we have to studied in
a special case false in the special case
we have studied if a plus B plus C is
equal to 0 so this a cube plus B cube
plus C cube will be equal to 3 ABC so
you see if you see here what we can see
here is if we consider this X minus y is
a x minus Z SB and z minus x SC and we
add all this thing that is X minus y
that is plus y minus Z plus Z minus X
will get the sum s 0 it means a plus B
plus C is equal to 0 as a plus B plus C
is equal to 0 so eq + b q plus C cube is
equal to 3 ABC it means X minus y whole
cube plus x minus at work across that
whereas x full cube is equal to 3 ABC
that is 3 into is X minus y b is y minus
Z and Z is Z minus X so that is equal to
3 ABC so in this way we can do the
preparation but instead of the cpa use
the energy that is a minus B with you we
got the same ultimate solution will be
very large and very complicated this is
not in then instead of that thing
whatever thing we have started we
actually have to visualize those
accordingly we need to solve the
question based on debt identity so this
is the way of doing the question or will
the characterization of by using the
identities so we will see for some more
example based on this
first we write down the question is in
this question what we need to do we need
actually have to find the value of find
the value of x cube minus eight x cube
minus 36 x y minus 2 16 when it is given
when x is equal to 2y plus 6 so in this
question we need to find out the value
of this expression where x is equal to i
plus 6 is given so what we can do is
first here is first will visualize this
question properly and accordingly you'll
see that a better we can use some
identity here for God because if you
directly keep the value of x is equal to
2y plus it this question will be very
complicated enlarge so instead of that
swill first understand what we can do
with this question is if you can see in
the first term is x cubed that is a cube
plus check in a second term is minus
eight x cube so we can write it as minus
two y whole cube so it will become plus
meet you and according in the same way
to 16 is actually the tube of six so we
can write the rest- six-fold cube so it
will become CQ and accordingly if he's
visualize the same i did you get it AQ +
b q + CQ minus three ABC so in this case
the minus three ABC will become minus 3
into is our X B is our minus two y and c
is our minus 6 so if you multiply all
this term will get minus 36 X Y so this
egg question is actually in the form of
eq + v qkc2 x the three ABC and be
enough to convert the pusher in that
form actually so most importantly we
have the first we have to visualize this
thing they're only be able to convert it
into the Volvo if you can speak you
critique you by the swing ABC so
accordingly this will be equal to a plus
B plus C into a square plus B Square C
square minus a b-minus bc- see it so we
can use directly that thing here but if
you can observe this question again
properly that it is given at x is equal
to 2y plus 6 and if we can on feet check
the
special case of the same identity
special case was e a plus B plus C is
equal to 0 this was that was the special
case so accordingly we have taken a s
here X instead of X we can keep 2y plus
6b here we have taken as minus 2y and
see here we are taken as minus 6 so if
you can add this thing will get 0 it
means we can use a special case of that
identity here that is if a plus B plus C
is equal to 0 it means this it means
that a cube plus b q plus C cube minus 3
ABC will be equal to 0 so value of this
expression is that could actually equal
to 0 because in a plus B plus C is zero
at rhs part of the same identity will be
equal to 0 VAR h is equal to 0 it means
the value of this expression will again
be equal to 0 so in this way we need to
understand which I need you to be used
where and accordingly we can solve the
question based on the identity and most
importantly the special case we have
studied of the identity QP q plus DQ
minus the ABC we have to remember that
special case records because it is very
your easy for them because it makes the
question very easy for us to solve
recording Lee get the right answer very
quickly