Examples based on Cube Identities - Polynomials Video Lecture - Class 9

so now we'll see some more example based
on the identity Rehan study first we
write down the example that is X plus 1
by X X plus 1 by X is equal to 7 so
accordingly what we need to do we need
to find out the value of x cube plus 1
by X Q so now what we can do is as X
plus 1 by X is given as seven so what we
can do is one method is would gladly
take the whole cube here so if you don't
take the whole cube air what we get that
is X plus 1 by y whole cube is equal to
7 q what we get X cube plus 1 by X cube
that is here using the identity a plus
the whole cube plus 3 AV that is 3 into
X into 1 by X 3 AV into a plus b rated X
plus 1 by X sewed it is it is the
identity for a plus B whole cube and
this is equal to 7 q get is 343 now what
we can do as we need to find out the
value of x cube plus 1 by X 2 so we have
already got X cube plus 1 by X cube now
what we can do is as you can c 3 s into
1 by XOXOX will get cancelled so we get
3 here and X plus 1 by X we know that I
o that is equal to 7 this is equal to 3
43 now again x cube plus 1 by x cube is
equal to 3 43 minus 21 and this is equal
to 3 22 so this is a value for x cube
plus 1 by ask you so this is the one
method we can you or there is one more
method like you can use identity for s
cube plus 1 bed cuba rocky and
accordingly we get the same answer so in
the same we need to understand like
bitch what vanity should be used there
and Delta that depends like how will be
our question and how we can solve it
easy now we move on to the next example
like next example if you can take in
this question it is given at a plus B is
equal to read it is given dead and a
square plus B Square
is equal to 58 so accordingly what we
need to do we need to find out the value
of a cube plus B cube so now what we can
do here is that as you need to find out
the value of a Q plus B cube now if you
see the identity for AQ + V Cube it is
better to use this a dare empty here
because we cannot use the same identity
which are used in the previous question
so as we can see that a cube plus b q is
equal to a plus b into a square minus a
B plus B Square this is the identity for
AQ + B cube as we know hot so in this
question that a plus b is given and a
square plus B squared is also given but
we don't know what is the value of a B
so to find out the value of av what we
can do we use the information that is
already given in the question that is a
plus B is equal to it and what we can do
here is we redo the whole square on both
the side while doing the whole square
what will get a square plus B Square +
to a B is equal to 64 and now here from
here we get the value of a B because is
purpose p square is already given that
is to AP is equal to 64 minus a squared
plus B squared at is 58 and if we solve
it what we get 6 so accordingly
therefore value of a B is equal to 6
divided by 2 that is 3 and now you keep
this value in the identity of a cube
plus B cube to find out the value of eq
+ B cube so a cube plus B cube is equal
to a plus B so a plus B as we know that
if it is equal to feed it into a square
plus b square minus 80 and a square plus
B Square as we know this 58 and minus a
be so it is equal to 3 now this is equal
to aid in 258 minus 3 is equal to 55
now if you solve it what we get here
accordingly if you solve this question
we get the value of a cube plus B cube
that is 55 into it so that will be equal
to 4 14 so this is the value of AQ + b
QP God so accordingly the question will
be solved on the basis of the identity
use because for a cuban v q AQ + V Cube
we can use two or three different
identities so it is very important to
understand which I need you to be used
now we move on to the next example so in
the next example if you see it is given
that X square plus 1 by x square is
equal to 7 so find out the value of x
cube plus 1 by X cube so how to solder
in the question it is given at the X
square plus 1 x square + 7 need to find
out the value of x2 + 1 x cube so what
we can do is as you know that identity
for X cube plus 1 by X cubed that is the
AQ + b q we can use a different entities
over at it is X plus 1 by x into x
square plus 1 by X square minus a B and
minus 80 here means minus X into 1 by x
that is equal to minus 1 now we know the
value of x square plus 1 by X square but
we do not know the value of x plus 1 by
X of what we can do to find out the
value of x plus 1 by s use the
information that is given that is X
square plus 1 by x square is equal to 7
and as you if you can see that s square
plus 1 by r square is X plus 1 by X
whole square minus 2 is equal to 7
because we can write x square plus 1 by
square s x + 1 x whole square minus 2
and this is equal to 7 so accordingly if
we solid what we get X plus 1 by X whole
square is equal to retain this minus 2
on the rhs part it will become plus 2
data 7 verse 29 when accordingly x
plus 1 by X is equal to under root 9
that is three so we caught the value of
x plus 1 by X and now we keep this value
therefore X cube plus 1 by X cube is
equal to X plus 1 by s that is three
next square plus 1 by x square is 7-1
and if you solid what will get here is
three you 26 DETT is 18 so this is the
answer we got 4 s cube plus 1 by ask you
know we'll see some more example based
on the identities next example is like
suppose if you need to find out the
value of 23 cube minus 17 cube so what
we can do there is one method like you
can hardly find out the value of
practically cubed 72 and subtracted will
get the value but if we know the
identities it will be easier for us to
solve this type of question because if
you see here it this is in the form of a
cube minus BQ they'd use the identity
that is a minus B or 23-17 into a square
that is 23 square plus B Square at is 17
square plus a B that is 23 into 17 so we
use this identity and 414 if you use
this is easier for us to solve this
question because 33-17 if you solid this
is equal to 6 accordingly 23 square 2 23
square as we all know that this is equal
to 5 39 + 17 square and 17 square is
equal to 2 89 + 23 into 17 so we'll get
the value of 23 into 17 and awkwardly if
we sorted it easier for us to solve it
so accordingly if you saw him we get the
answer is 725 food so then let in some
numbers or in cases of number knows that
we use the identity of what areas solve
the question so it will be easier this I
didn't make this this type of question
very easier because finding the 23 Cuban
accordingly 17 cube and then taking the
difference
is little bit more complicated and
lengthy and soldering using identities
is easier for us understand and also to
find out the value very quickly now we
move on to the next example like suppose
is a plus B is equal to 7 it is given
and a B is equal to 2n it is also given
and if you need to find out the value of
a square minus a B plus B Square so how
to find out the value of a square minus
80 plus B Square not so now as you can
see we get a square plus b square minus
a be you know the value of a me but we
don't know the value of a square plus B
Square and we use the very use the
information that is given in the
question to find out the value of a
square plus B Square now if you can see
a plus B is equal to 7 and if you do the
whole square here but what will get here
is a square plus B Square + to a B is
equal to 49 so now if you solve it what
we get is a square plus b square is
equal to 49 minus 2 80 and 80 is given
as 2 n which is equal to 2 into 12 so
that is 49 minus 24 so this is equal to
25 and now we keep this value here that
is a square minus kv + v square so as we
noted v squared plus b squared is 25 and
minus KP is well so accordingly the
answer for the e square minus 80 plus b
square is study so this is the matter
this is the way to solve the questions
so now you see some more example based
on the different identities let us write
on the example let us take the example
as if a plus B plus C is equal to 6 it
is given end a B plus BC plus CA is
equal to 11 it is given so accordingly
what we need to do is we need to
out the value of a cube plus B cube plus
C cube minus 3 epc so we need to find
out the value of this so how to solid is
so if you can observe this question if a
plus B plus C is equal to such an AP
plus vg+ c is equal to 11 so accordingly
we need to find out the value so
according to use the identity which has
studied military to eq + PQ + CQ minus 3
ABC so if you see here that is a cube
plus B cube plus C cube minus 3 ABC so
this is equal to if you see the identity
remember I did it is equal to a plus B
plus C into a square plus B Square the C
square minus a V minus VC minus C so in
this question if we know the value of a
plus B plus C you know the value of
maybe B's plus BC plus CL so but we do
not know the value of a square plus B
squared plus C square so what we can do
is because the information that is given
yes a plus B plus C is equal to 6 and if
you do the whole square on both the side
what will get a square plus B Square + C
square plus 2 eb + 2 b.c the stew CA is
equal to 36 and from this we get the
value of a square plus b square plus z
square so a squared plus b square plus C
square is equal to 36 minus if you take
the common to as common from to able to
be easier to CL we get in the bracket ad
plus BC plus see him so this is equal to
36 minus 2 into 80 + b C + CN so a B
plus BC plus CN is equal to 11 so this
is equal to 36 minus 22 and which is
equal to 14 so this is the value of a
square plus B Square + C square and now
what we can do we need to find out the
value of
AQ + B cube plus C Q minus V ABC so
we'll keep the value in the identity
that is a plus B plus C and it was B
plus C is given as 6 into a square plus
B squared plus z square which is equal
to 14 minus AP minus VCC minus here so
among minus 80 minus minus VC minus T if
you take the former mind has what we get
maybe + b C plus D and value of that
name p plus BC plus c is equal to 11 so
now if a solid red districts into 14-11
is 3 that is equal to 18 so this is the
value of a cube cosmic you persecute
minus readings so in this way we need to
understand the identity accordingly
under center position and then we'll be
able to solve the question properly now
we'll see one more example based on this
this first right of the example as what
we need to do is evaluate minus 12 q + 7
q + 5 cube that is we need to find out
the value of this so if we solve this
question by taking the cube of all the
three numbered individually and then
adding those numbers it will be little
complicated instead of this this what we
can do here is we have studied the
identity that is a cube plus DQ plus C
cube minus a cube plus make u plus CT
2-3 ABC is equal to a plus B plus C into
a square plus B Square + C square minus
ei be minus VC minus C so now in this
identity has here studied that if a plus
B plus C is equal to 0 it means our
whole RHS part of this identity will
become 0 as a plus B plus C is equal to
0 in that case AQ + b q + CQ is will
become equal to 3 ABC so in the question
it is given gratis need to find out
where you
12 on q + 7 q + 5 you so in this
question a will be equal to minus 12 v
will be equal to 7 and c will be equal
to 5 and if we take find out the value
of a plus B plus C that is minus 2l plus
7 plus 5 it is coming as zero so it
means the value of minus 12 you or well
or if a plus B plus C is equal to 0 it
means a cube plus EQ the C cube is equal
to 3 ABC so if me is minus 2 lq + 7 cube
plus file you will be equal to 3 into
minus 12 in 27 in 25 and accordingly if
we solve it what will get the value will
get the renewals minus 1 through 6 0 so
now this question becomes easier to
solve the solid as is the entity here
before but if you have not user identity
it will be it is more complicated and
lengthy so accordingly some put this
identity is also very useful enough what
a thing is is to remember the special
case of the identity if you remember the
special case it will be easier for us to
solve the question and accordingly get
the answer now next question is what we
need to do is we need to simplify that
is a square minus B Square whole cube
plus b square minus C square whole cube
plus C square minus a square whole cube
divided by a minus B whole cube plus B
minus C 1 cube plus C minus a whole cube
so accordingly what we need to do we
need to simplify this so now if in this
question if you see yet it is actually
in the form of lubricating amador
actually in the form of a minus B whole
cube a minus B want you so if you use
those that identity this question will
become more complicated and very lengthy
because in their identity we need to use
three times the new
because there are three terms with with
the hydrant ed rat is a minus B hole too
accordingly b square minus a square what
he was again in the form of e minus q
whole cube so according you need to use
the same identity three times in the
numerator and three times in the
denominator so this question will become
very lengthy apart from this what we can
do is if you see here it is a square
minus b square b square minus a square
and c square minus a square that is if
we consider this whole term as a or
let's say if you we know the identity at
is a cube plus b cubed hasta q minus 3
NB c is equal to so and so so instead
and in that case is a plus b plus c is
equal to 0 SE eq + eq + C cube is equal
to 3 so instead of dead also we can use
s cube plus y cube plus Z cube is equal
to three XYZ because a and B already
used in the question so if you take a
square minus b square as x b square
minus c square as y and c square minus a
square head z and if you do take the sum
of this thing that is X plus y plus Z it
is coming as zero because a square minus
b square plus b square minus a square
plus C square minus a square if you saw
it will get it is as zero it means X
plus y plus Z is equal to 0 with this
question and as X plus y plus Z is equal
to 0 s cube plus y q / z 2 will be equal
to 3x why I said it means that is a
square minus B Square whole cube plus b
square minus C square vole cube plus C
square minus a square whole cube will be
equal to 3 times 0 a B and C and three
times of a B and C means three times of
a square minus b square b square minus a
square and c square minus a square so
this thing be caught in the numerator
and same as the denominator denomination
what is the same thing now in this case
we take this a minus vs
next you be- csv and c minus SW and if
you take the sum of this u plus V plus W
it is coming at coming as zero it means
that is a minus B whole cube plus B
minus C whole cube plus C minus a whole
cube will be equal to three times of UV
NW dirty three times of a minus B B
minus C and C minus a so this thing we
got in the denominator and now if you
divide it because after what we need to
do is we need to find out an area of a
square minus B Square whole cube plus v
square minus C square whole cube plus C
square minus a square hole view / a
minus B whole cube plus B minus C whole
cube when plus C minus a whole cube and
now as we got the value of e square my
numerator that is three times of e
squared minus B Square so we write it as
a minus b to e plus b square minus B
Square a particle edu square minus a
square antennas B minus C n into B plus
C when same way c minus a and c plus a
so this will become our numerator in the
same way write the denominator value of
denominator is 3 times of e minus B B
minus C and C minus a which we got
previously in the question so now if we
divided 3 and fig attention u- b a- big
Atkinson V minus c u minus e get
considered c minus n c minus a will get
cancelled and now the answer me god has
a plus b into b plus c into c plus e so
this is the value we got while solving
this question