kf7
: 4.1
dxTtd ;dfjt{s / n3'Ttd ;dfjt{s -d=;= / n=;=_
(Highest Common Factor and Lowest Common Multiple)
1=
kl/ro
lbOPsf b'O{ jf b'O{eGbf a9L aLh ul0ftLo cleJo~hsx¿sf] ;femf u'0fg v08 jf ;femf u'0fg
v08x¿sf] u'0fgkmnnfO{ d=;= (H.C.F) elgG5 eg] b'O{ jf b'O{eGbf a9L aLh ul0ftLo
cleJo~hsx¿n] lgz]if efu hfg] ;a}eGbf ;fgf] cleJo~hsnfO{ n=;= (L.C.M.) elgG5 . o;
PsfOdf aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 d=;= / n=;= lgsfNg] tl/sfx¿sf
af/]df rrf{ ul/g] 5 .
o; kf7af6 nfdf] pTt/ cfpg] Pp6f ;d:of ;dfwfg txsf] k|Zg ;f]lwg] 5 . h;sf] cª\s ef/
/xg] 5 .
2.
o; kf7df k|of]u x'g] cfwf/e"t ;"qx¿
-s_ d=;=
-v_ n=;=
-u_
(H.C.F)
= ;femf u'0fg v08x¿ (common factors)
(L.C.M) = ;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors)
n=;= × d=;= = klxnf] cleJo~hs × bf];|f] cleJo~hs
L.C.M × H.C.F = First Expression × Second Expression
-3_ aLh ul0ftLo ;"qx¿
-c_ (a + b)2 = a2 + 2ab + b2
-cf_ (a – b)2 = a2 – 2ab + b2
or, (a + b)2 = (a – b)2 + 4ab
or (a + b)2 – 4ab = (a – b)2
a2 – b2 = (a + b) (a – b)
-O_
-O{_ a2 + b2 = (a + b)2 – 2ab = (a – b)2 + 2ab
-p_ a3 + b3 = (a + b) (a2 – ab + b2) = (a + b)3 – 3ab (a+ b)
-pm_ a3 – b3 = (a – b) (a2 + ab + b2) = (a – b)3 + 3ab (a–b)
-P_ (a + b)3 = a3 + 3a2b + 3ab2 + b3
-P]_ (a – b)3 = a3 – 3a2b + 3ab2 – b3
-cf]_ (a + b + c)2 = a2 + b2 + c2+ 2ab + 2bc + 2ca
-cf}_ a3 + b3 + c3– 3abc =(a + b + c) (a2 + b2 + c2 – ab – bc – ca)
3. ljz]if Wofg lbg'kg{] s'/fx¿
o; PsfOdf
1. aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 d=;= lgsfNg'kg{] x'G5 .
2. aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 n=;= kTtf nufpg'kg{] x'G5 .
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
115
4
3.
4.
5.
–
–
–
6.
lbOPsf b'O{ jf ltg cf]6f dfq aLh ul0ftLo cleJo~hsx¿sf] d=;=/ n=;= lgsfNg'kg{]
x'G5 .
lbOPsf cleJo~hsx¿sf] d=;= eGgfn] ;a} cleJo~hsx¿sf] ;femf u'0fg v08 jf u'0fg
v08x¿sf] u'0fg kmn xf] .
d=;= lgsfNbf,
;a} cleJo~hsx¿sf] v08Ls/0f ug]{,
tL cleJo~hsx¿sf] ;femf u'0fg v08x¿ lng]
;femf u'0fg v08x¿nfO{ u'0fgkmnsf] ¿k (product form) df n]Vg]
n=;= eGgfn] b'O{ jf b'O{eGbf a9L aLh ul0ftLo cleJo~hsx¿sf] ;femf u'0fg v08 × af“sL
u'0fg v08nfO{ a'emfp“5 .
4.
gd'gf k|Zgf]Tt/ tyf cEof;
gd'gf 1 :
d=;= lgsfNg'xf];\ M (Find the H.C.F)
x 3 - 64y3, x 2 - 6x y + 8y2 and x 2 - 16y2
;dfwfg
oxfF, klxnf] cleJo~hs
: x 3 - 64y3
= x 3 – (4y)3
= (x -4y){x2+ xx 4y+(4y)2}
= (x -4y)(x2+ 4xy+16y2)
bf];|f] cleJo~hs : x 2 - 6xy + 8y2
= x 2 – (4+2) xy + 8y2
= x 2 – 4xy - 2xy + 8y2
= x(x - 4y) – 2y(x – 4y)
= (x - 4y) (x – 2y)
t];|f] cleJo~hs : x 2 - 16y2
= x 2 – (4y)2
d=;=
1.
(a)
(b)
(c)
(d)
( H.C.F) =
= (x – 4y)(x + 4y)
;femf u'0fg v08x¿
= (x - 4y)
cEof;sf nflu k|Zgx?
(common factors)
4x 2 - 9, 2x 2 + x - 3 and 8x 3 + 27
3p2 - 9pq - 12q2, 4p2 - 18pq + 8q2 & 3p2 - 18pq + 24q2
x 3 - 9x y2, 2x 2 + 11x y + 15y2 &x 3 + 27y3
4x 4 + 16x 3 - 20x 2, 3x 3 + 14x 2 - 5x & x 4 +125x
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
116
(e)
(f)
(g)
(h)
16x3y - 4 x 2y2 - 30x y3, 32x 4 - 50x 2y2 & 4x 2y2 + 9x y3 + 5y4
(a + 2b)2, (a -2b)2 + 8ab & a4 - 16b4
3t4 - 8t3 + 4t2, t5 - 8t2 & 4t3 - 10t2 + 4t
36 x 4 - 81x 2, 24x 3 - 72x 2 + 54x & 24x 6 - 81x 3
pTt/x¿
(a) (2x +3), (b) p - 4q, (c)
(g) t (t - 2), (h) 3
gd'gf 2 :
d=; lgsfNg'xf];\ M
x + 3y, (d) x (x + 5), (e) 4 x + 5y, (f) a + 2b
x (2 x - 3)
(Find the H.C.F)
x + x y + y , x 4 + x 3y + x2y2 and x4 - xy3
4
2 2
4
;dfwfg M
oxfF, klxnf] cleJo~hs M x 4 + x2y2 + y4
= (x 2)2 + (y2)2 +
x 2y2
= (x 2 + y2)2 – 2x2y2+
= (x + y ) – x y
2
2 2
2 2
x 2y2
= (x 2 + y2)2 – (xy)2
= (x 2 + y2 + xy) (x 2 + y2 - x y)
= (x 2 +xy+ y2 ) (x 2 - x y + y2 )
bf];|f] cleJo~hs M
x 4 + x 3y + x 2y2
= x 2 (x 2 + x y + y2)
t];|f] cleJo~hs M x4 - xy3
= x (x3 – y3)
d=;=
= x (x – y) (x 2+ xy+y2)
( H.C.F) = ;femf
u'0fg v08x¿
= (x + x y + y )
2
2
(common factors)
cEof;sf nflu k|Zgx?
(a)
(b)
(c)
(d)
(e)
(f)
8a3 + b3, 16a4 + 4a2b2 + b4
x3 - 8y3, x4 + 4x 2y2 + 16y4
a4 + 4a2 + 16, a3 + 8
a4 + a2 + 1, a3 - 1
x 4 + 4x2 + 16, x 3 - 8
m4 + m2n2 + n4, m2n3 + m5
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
117
(g)
(h)
(i)
4x4 + 19x 2y2 + 49y4, 4x3 + 14x y2 - 6
x 2y
x 3 - 1, x4 + x 2 + 1, x3 + 1 + 2x2 + 2x
4x 4 +16x 3 - 20x 2, 3x3 + 14x 2 - 5x and x 4 + 125x
pTt/x¿
(a) 4a2 - 2ab + b2, (b) x 2 + 2 x y + 4y2, (c) a2 - 2a + 4, (d) a2 + a + 1, (e) x 2 + 2 x + 4,
(f) m2 + mn +n2, (g) 2x 2 - 3xy +7y2, (h) x 2 + x +1, (i) x (x+5)
gd'gf 3 :
d=;= lgsfNg'xf];\
(Find the H.C.F)
x 3 - 2x 2 - x + 2, x 3 - 3x 2 + 2x and x 2 - x - 2
;dfwfg M
oxfF, klxnf] cleJo~hs M x3 - 2x 2 - x + 2
= x 2(x – 2) - 1(x – 2)
= (x – 2) (x 2 – 1)
= (x – 2) (x 2 – 12)
= (x – 2) (x + 1) (x – 1)
x 3 - 3x 2 + 2 x
= x (x 2 - 3x + 2)
= x {x 2 - (2+1) x+2}
= x {x 2 – 2x – x +2}
= x { x (x – 2) – 1(x – 2)}
= x (x – 2) (x – 1)
t];|f] cleJo~hs M
x2 - x - 2
= x 2 – (2-1) x - 2
= x2 – 2 x + x - 2
= x (x – 2) + 1(x – 2)
= (x – 2) (x + 1)
bf];|f] cleJo~hs M
d=;=
( H.C.F) =
;femf u'0fg v08x¿
= (x – 2)
(common factors)
yk cEof;sf nflu k|Zgx?
d=;= lgsfNg'xf];\ M (Find the H.C.F)
(a) a3 - a2 + a - 1 / 2a3 - a2 + a - 2
x 3 - x 2 + x - 1 / 2x 3 - x 2 + x - 2
(c) p3+1, p4+p2+1 / p3+1+2p2+2p
(d) 8a3 + 1 / 16a4 - 4a2 + 4a - 1
(b)
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
118
(e) 8x 3 - 1 / 16x 4 - 4x 2 - 4
x-1
(f) 2x 3 - 16, x 2 - 4x + 4 / x 2 - 3x + 2
pTt/x?
(a) a - 1, (b) x – 1, (c) 1, (d) 4a2 - 2a + 1, (e) 4x 2 + 2x + 1, (f) x - 2
gd'gf 4 :
d=;= lgsfNg'xf];\ M
(Find the H.C.F)
x - y + z + 2x z, x 2 + y2 - z2 + 2xy, z2 + y2 - x 2 + 2yz
2
2
2
;dfwfg M
oxfF, klxnf] cleJo~hs
:
x 2 - y2 + z2 + 2xz
= x 2 + 2x z+ z2 - y2
= (x + z)2 – y2
= (x + z + y) (x + z - y)
bf];|f] cleJo~hs
t];|f] cleJo~hs
= (x + y + z) (x + z - y)
x 2 + y2 - z2 + 2 x y
= x 2 + 2xy + y2 - z2
= (x + y)2 – z2
= (x + y + z) (x + y - z)
:
: z2 + y2 - x2 + 2yz
= y2+ 2yz + z2 - x2
= (y + z)2 – x 2
= (y + z + x) (y + z - x)
d=;=
= (x + y + z) (y + z - x)
( H.C.F) = ;femf
u'0fg v08x¿
= (x + y + z)
(common factors)
cEof;sf nflu k|Zgx?
d=; lgsfNg'xf];\ M (Find the H.C.F)
(a) a2 + 2ab + b2 - c2, b2 + c2 - a2 + 2bc, c2 + a2 + 2ca - b2
(b) e2 + f2 - g2 + 2ef, e2 - f2 + g2 + 2ge, g2 + f2 + 2fg - e2
(c) 9m2 - 4n2 - 4nr - r2, r2 - 4n2 - 9m2 - 12mn and 9m2 + 6mr + r2 - 4n2
pTt/x¿
(a) a + b+ c, (b) e + f + g, (c) 3m + 2n + r
gd'gf 5 :
n=;= lgsfNg'xf];\ (Find the L.C.M.)
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
119
a5 + a3b2 + ab4 and a4b + ab4
;dfwfg M
oxfF, klxnf] cleJo~hs
: a5 + a3b2 + ab4
= a{(a ) + (b2)2+ a2b2}
=a{ (a2)2 + (b2)2+ a2b2}
= a{(a2 + b2)2 - 2a2b2 + a2b2}
= a{(a2 + b2)2 – (ab)2}
= a (a2 + b2 + ab) (a2 + b2 – ab)
bf];|f] cleJo~hs M a4b + ab4
= ab(a3 – b3)
= ab(a – b) (a2 + ab + b2)
n=;= (L.C.M.) = ;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors)
= ab (a – b) (a2 + ab + b2) (a2 + b2 – ab)
2 2
cEof;sf nflu k|Zgx?M
n=;= lgsfNg'xf];\ (Find the L.C.M.)
(a) x 4 + x 2y2 + y4 & x 4 - xy3
(b) 6x 2 - x - 1 & 54 x 4 + 2x
(c) (2x 2 - x - 1) & 24x4 + 3x
(d) x 3 - 3x 2 - x + 3 & x 3 - x 2 - 9x + 9
(e) x 3 + 2x 2 - x - 2 & x3 + x 2 - 4x - 4
(f) x 3 - 9x, x 4 - 2x3 - 3x 2 & x3 - 27
(g) a3 - 4a, a4 + a3 - 2a2 and a3 - 8
pTt/x¿
x (x -y) (x 2 + x y + y2) (x 2 - x y + y2)
(b) 2x (3x +1) (2x -1) (9x 2-3x +1)
(c) 3x (x -1), (2x +1)(4x 2 - 2x + 1)
(d) (x + 3) (x + 1) (x - 1) (x - 3)
(e) (x - 1) (x - 2) (x +1) (x + 2)
(f) x 2 (x + 3) (x 3 - 27) (x + 1)
(a)
(g) a2(a2 - 4) ( a2 + 2a + 4)(a - 1)
gd'gfM 6 :
d=;= / n=;= lgsfNg'xf];\ (Find the H.C.F. and L.C.M.)
2(x 2 - y2), 4(x 3 - y3)
/ 6(x 4 - y4)
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
120
;dfwfg M
oxfF, klxnf] cleJo~hs
: 2(x 2 - y2),
bf];|f] cleJo~hs M
4(x 3 - y3)
= 2(x + y) (x - y)
= 2×2(x - y) (x 2 + xy + y2)
6(x 4 - y4)
t];|f] cleJo~hs M
= 6{(x 2)2 – (y2)2}
= 6(x 2 + y2) (x 2 – y2)
= 2×3(x 2 + y2) (x + y)(
d=;. ( H.C.F) = ;femf u'0fg v08x¿
n=;=
(L.C.M.) =
= 2(x – y)
x – y)
(common factors)
;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors )
= 2(x – y)(x + y)2×3(x 2 + y2) (x 2 + x y + y2)
= 12(x – y)(x + y)(x 2 + y2) (x2 + xy + y2)
= 12(x 2– y2)(x 2 + y2) (x 2 + x y + y2)
cEof;sf nflu k|Zgx?
d=;= / n=;= lgsfNg'xf];\
(Find the H.C.F. and L.C.M.)
x - 1, (x - 1) , x + x + 1
(b) x 3y + y4, x 4 + x 2y2 + y4, 2ax 3 - 2ax 2y + 2axy2
(c) 2 x 3 - x 2 - x, 4x 4 - 10x 3 - 6x 2, 8x 2 + 4x
(d) 8a6x - 8a3 x 4, 4a6 x 2 - 4a2x 6, 6x 3a3 + 24a2x 4 - 30ax 5
(a)
(e)
(f)
3
3
4
2
3a3b + a2b2 - 10ab3, 6a4b - a3b2 - 15a2b3, 6a3b - 19a2b2+15ab3
x2 - 2xy + y2, 2x2 + xy - 3y2, x3 - 3x2y + 3xy2 - y3
pTt/x?
(a)
(b)
(c)
(d)
(e)
(f)
H.C.F = 1, L.C.M. = (x - 1)3 ( x2 + x + 1) (x2 - x +1)
H.C.F. = x2 - xy +y2, L.C.M = 2axy ( x+ y) (x2 - xy + y2) (x2 + xy + y2)
H.C.F.= x(2x+1), L.C.M.= (4x2 (2x +1) ( x- 1) (x - 3)
H.C.F. = 2ax ( a - x), L.C.M.= 24a3x3 ( a - x) (a + x) (a2 + x2) (a + 5x) (a2 + ax + x2)
H.C.F. = ab(3a - 5b), L.C.M.= a2b(3a - 5b) (a + 2b) (2a + 3b) (2a - 3b)
H.C.F.= x - y, L.C.M.= (x- y)3 ( 2x + 3y)
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
121
yk cEof;sf nflu k|Zgx? (Challenging Problems)
d=;= lgsfNg'xf];\ M (Find the H.C.F.)
(a) x4 + 2x2 + 9 & x3 - x + 6
(b) x3 - 3x - 2 & x3 - x2 - 4
(c) a4 - 6a2 - 7 + 8x - x2 & a3 - ax + a
(d) x4 - 10x2 + 24 + 6y - 9y2 & x3 + 3xy - 6x
Hints (d):
;dfwfg M
oxfF, klxnf] cleJo~hs
pTt/x¿
(a) x 2 - 2x + 3
(d) x 2 + 3y - 6
:
=
=
=
=
x 4 - 10 x 2 + 24 + 6y - 9y2
(x 2)2 - 2. x 2.5 + 52 - 1 + 6y - 9y2
(x 2 - 5)2 - [12-2.1.3y + (3y)2
(x 2 - 5)2 - ( 1- 3y)2
(x 2 + 3y-6) (x 2 - 3y - 4)
(b) x - 2
(c) (a2 - x + 1)
yk cEof;sf nflu k|Zgx?
d=;= / n=;= lgsfNg'xf];\ (Find the H.C.F. and L.C.M.)
(a)
(b)
(c)
(d)
(e)
pTt/x?
(a)
(b)
(c)
(d)
(e)
x 2 + 2x - 3, x 3 + 3x 2 - x - 3
x 2 - 4x + 3, x 3 - x 2 - 14x + 24
3x 2 + 16x - 35, x3 + 343, 2x 2 - 9x - 161
x 3 - 3x + 2, x 3 + 4x 2 - 8x + 3
3x 2 - 22 x + 19, 2x 3 + 3x - 5
H.C.F. = (x - 1) (x + 3), L.C.M.= (x - 1) (x + 3) (x + 1)
H.C.F.= x - 3, L.C.M.= (x - 1) (x - 2) (x - 3) (x + 4)
H.C.F. = (x + 7), L.C.M = (x + 7) (3x - 5) (2x - 23) (x 2 - 7x + 49)
H.C.F. = (x - 1), L.C.M.= (x - 1)2 (x + 2) (x2 + 5x - 3)
H.C.F. = (x - 1), L.C.M.= (x - 1) (3x - 19) (2x 2 + 2x + 5)
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
122