次の式を因数分解しなさい。
1 6a x y - 15ax y + 9axy
( ) 2 3 2 2 2 ( ) (2 x - y - x + y)2
3 x - 27
( ) 3 ( )4 3x + 5x + 22
5 4p - 12pq + 9q
( ) 2 2 ( )6 x - 2y - xy - 2x - 5y - 32 2
7 2 x - 1 - 7 x - 1 + 6
( ) ( )2 ( ) ( )8 6x y - 24xy3 3
9 3a - 6ab - 11a + 4b + 6
( ) 2 ( )10 a + 6a + 12a + 83 2
11 x + 4x + 16
( ) 4 2 ( )12 10x + 11xy - 6y2 2
13 a b + ab + b c + bc + c a + ca + 2abc
( ) 2 2 2 2 2 2
= x - y x - y - 1( )( ) 3 x - 27 = x - 3 (x + 3x + 9
( ) 3 ( ) 2 ) ( )4 3x + 5x + 2 = x + 1 3x + 22 ( )( ) 5 4p - 12pq + 9q = 2p - 3q
( ) 2 2 ( )2 ( )6 x - 2y - xy - 2x - 5y - 32 2
= x - y + 2 x - 2y - 5y - 32 ( ) 2
= x - y + 2 x - (2y + 5y + 32 ( ) 2 )
= x - y + 2 x - (y + 1 (2y + 32 ( ) ) )
= x + y + 1 } x - 2y + 3 }{ ( ) { ( )
= x + y + 1 x - 2y - 3( )( ) 7 2 x - 1 - 7 x - 1 + 6
( ) ( )2 ( )
= 2A - 7A + 62
= A - 2 2A - 3( )( )
= x - 1 - 2 2x - 2 - 3( )( )
= x - 3 2x - 5( )( )
8 6x y - 24xy
( ) 3 3
= 6xy(x - 4y2 2)
= 6xy x + 2y x - 2y( )( )
9 3a - 6ab - 11a + 4b + 6 ( ) 2
= 4 - 6a b + 3a - 11a + 6( ) 2
= -2 3a - 2 b + a - 3 3a - 2 ( ) ( )( )
= 3a - 2 a - 2b - 3( )( )
10 a + 6a + 12a + 8 = a + 2
( ) 3 2 ( )3
11 x + 4x + 16
( ) 4 2
= x + 8x + 16 - 4x4 2 2
= (x + 4 - 4x2 )2 2
= (x + 4) + 2x} (x + 4) - 2x}{ 2 { 2
= (x + 2x + 4 (x - 2x + 42 ) 2 )
12 10x + 11xy - 6y = 2x + 3y 5x - 2y
( ) 2 2 ( )( )
13 a b + ab + b c + bc + c a + ca + 2abc
( ) 2 2 2 2 2 2
= b + c a + (b + 2bc + c )a + b c + bc( ) 2 2 2 2 2
= b + c a + b + c a + bc b + c( ) 2 ( )2 ( )
= b + c a + b + c a + bc}( ){ 2 ( )
= b + c a + b a + c( )( )( )
= a + b b + c c + a( )( )( )
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次の式を因数分解しなさい。
1 x + x - 4x - 4
( ) 3 2 ( )2 6a + 5ab - 4b2 2
3 12a - 16a - 3
( ) 2 ( )4 8a + b3 3
5 6x - 17xy + 12y
( ) 2 2 ( )6 (a + 3a - 2 a + 3a - 82 )2 2
7 a - b y + bx - ax
( ) ( ) ( )8 1 + 3ab + a + 3b
9 a - a b - ac + bc
( ) 3 2 2 2 ( )10 a b - c + b c - a + c a - b2( ) 2( ) 2( ) 11 2a - 3ab - 2b - a + 7b - 3
( ) 2 2 ( )12 x - 7x + 124 2
13 a + 3a + 4
( ) 4 2
= x + 1 (x - 4( ) 2 )
= x + 1 x + 2 x - 2( )( )( )
3 12a - 16a - 3 = 2a - 3 6a + 1
( ) 2 ( )( ) ( )4 8a + b = 2a + b (4a - 2ab + b3 3 ( ) 2 2) 5 6x - 17xy + 12y = 2x - 3y 3x - 4y
( ) 2 2 ( )( ) ( )6 (a + 3a - 2 a + 3a - 82 )2 2
= A - 2A - 82
= A + 2 A - 4( )( )
= (a + 3a + 2 (a + 3a - 42 ) 2 )
= a + 1 a + 2 a + 4 a - 1( )( )( )( ) 7 a - b y + bx - ax
( ) ( )
= a - b y + b - a x( ) ( )
= a - b y - a - b x( ) ( )
= a - b y - x( )( )
8 1 + 3ab + a + 3b ( )= 3b + 1 a + 3b + 1( )
= 3b + 1 a + 1( )( )
9 a - a b - ac + bc
( ) 3 2 2 2
= (c - a b + a - ac2 2) 3 2
= (c - a b + a(a - c2 2) 2 2)
= (c - a b - a2 2)( )
= c + a c - a b - a( )( )( )
= a - b a - c a + c( )( )( )
10 a b - c + b c - a + c a - b
( ) 2( ) 2( ) 2( )
= b - c a - (b - c a + b c - bc( ) 2 2 2) 2 2
= b - c a - b + c b - c a + bc b - c( ) 2 ( )( ) ( )
= b - c a - b + c a + bc}( ){ 2 ( )
= b - c a - b a - c( )( )( )
= a - b b - c a - c( )( )( ) 11 2a - 3ab - 2b - a + 7b - 3
( ) 2 2
= 2a - 3b + 1 a - (2b - 7b + 32 ( ) 2 )
= 2a - 3b + 1 a - b - 3 2b - 1 2 ( ) ( )( )
= a - 2b - 1 } 2a + b - 3 }{ ( ) { ( )
= a - 2b + 1 2a + b - 3( )( )
12 x - 7x + 12
( ) 4 2
= (x - 4 (x - 32 ) 2 )
= x + 2 x - 2 (x - 3( )( ) 2 )
13 a + 3a + 4
( ) 4 2
= a + 4a + 4 - a4 2 2
= (a + 2 - a2 )2 2
= (a + 2) + a} (a + 2) - a}{ 2 { 2
= (a + a + 2 (a - a + 22 ) 2 )
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次の式を因数分解しなさい。
1 a x - y - by + bx
( ) ( ) ( )2 a + 27b3 3
3 6x + x - 15
( ) 2 ( )4 x + 2y - 3xy + 5x - 8y + 62 2
5 4x - 8x - x + 2
( ) 3 2 ( )6 x + 2x + 94 2
7 2 + 2ab - 4a - b
( ) ( )8 8x + 2xy - 15y2 2
9 a + 4a - 5
( ) 4 2 ( )10 2abc - a b - b c - a c - bc + ab + ac2 2 2 2 2 2 11 3x + 10xy - 8y
( ) 2 2 ( )12 b - ab - bc + ac3 2 2 2
13 (x + 5x + 10(x + 5x + 24
( ) 2 )2 2 ) ( )14 a - 3a + 3a - 13 2
= a + b x - y( )( )
3 6x + x - 15 = 2x - 3 3x + 5
( ) 2 ( )( ) ( )4 x + 2y - 3xy + 5x - 8y + 62 2
= x - 3y - 5 x + 2y - 8y + 62 ( ) 2
= x - 3y - 5 x + 2(y - 4y + 32 ( ) 2 )
= x - 3y - 5 x + 2 y - 1 y - 32 ( ) ( )( )
= x - 2 y - 1 x - y - 3{ ( )}{ ( )}
= x - 2y + 2 x - y + 3( )( ) 5 4x - 8x - x + 2
( ) 3 2
= 4x x - 2 - x - 22( ) ( )
= x - 2 (4x - 1( ) 2 )
= x - 2 2x + 1 (2x - 1( )( ) )
6 x + 2x + 9
( ) 4 2
= x + 6x + 9 - 4x4 2 2
= (x + 3 - 4x2 )2 2
= (x + 3 + 2x (x + 3 - 2x2 ) 2 )
= (x + 2x + 3 (x - 2x + 32 ) 2 ) 7 2 + 2ab - 4a - b
( )= 2ab - 4a - b + 2
= 2a b - 2 - b - 2( ) ( )
= 2a - 1 b - 2( )( )
8 8x + 2xy - 15y
( ) 2 2
= 2x + 3y 4x - 5y( )( )
9 a + 4a - 5
( ) 4 2
= (a - 1 (a + 52 ) 2 )
= a + 1 a - 1 (a + 5( )( ) 2 )
10 2abc - a b - b c - a c - bc + ab + ac
( ) 2 2 2 2 2 2
= -a b - a c + 2abc + ab + ac - b c - bc2 2 2 2 2 2
= -(a b + a c - 2abc - ab - ac + b c + bc2 2 2 2 2 2)
= - b + c a - (b + 2bc + c )a + bc b + c( ) 2 2 2 ( )
= - b + c a - (b + c) a + bc b + c( ) 2 2 ( )
= - b + c a - (b + c)a + bc( ) 2
= - b + c a - b a - c( )( )( )
= a - b b + c c - a( )( )( ) 11 3x + 10xy - 8y
( ) 2 2
= x + 4y 3x - 2y( )( ) ( )12 b - ab - bc + ac3 2 2 2
= ac - ab + b - bc2 2 3 2
= a(c - b + b(b - c2 2) 2 2)
= a c + b c - b - b c + b c - b( )( ) ( )( )
= a - b b + c c - b( )( )( )
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= (x + 5x + 4 (x + 5x + 62 ) 2 )
= x + 1 x + 4 x + 2 x + 3( )( )( )( )
= x + 1 x + 2 x + 3 x + 4( )( )( )( )
1 6x + xy - y
( ) 2 2 ( )2 2x - x - 8x + 43 2
3 2x - 2y + 3xy + 11x - 3y + 5
( ) 2 2 ( )4 6a + 23ab - 48b2 2
5 2a - 16b
( ) 3 3 ( )6 x + 4x + 164 2
7 2a + 3b - 6ab - 1
( ) ( )8 5x + 8x - 42
9 x - 6x + 12x - 8
( ) 3 2 ( )10 x - xy - x - y - 22
11 (x + y - (x - y
( ) )4 )4 ( )12 4x + 14
13 x + 1 x + 2 x + 3 x + 4 - 24
( ) ( )( )( )( ) ( )14 abc + ab + bc + ca + a + b + c + 1
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1 6x + xy - y = 2x + y 3x - y
( ) 2 2 ( )( ) ( )2 2x - x - 8x + 43 2
= x 2x - 1 - 4 2x - 12( ) ( )
= 2x - 1 (x - 4( ) 2 )
= 2x - 1 x + 2 x - 2( )( )( ) 3 2x - 2y + 3xy + 11x - 3y + 5
( ) 2 2
= 2x + 3y + 11 x - 2y - 3y + 52 ( ) 2
= 2x + 3y + 11 x - (2y + 3y - 52 ( ) 2 )
= 2x + 3y + 11 x - (y - 1 2y + 52 ( ) )( )
= x + 2y + 5 2x - y - 1{ ( )}{ ( )}
= x + 2y + 5 2x - y + 1( )( )
4 6a + 23ab - 48b = 2a - 3b 3a + 16b
( ) 2 2 ( )( )
5 2a - 16b
( ) 3 3
= 2(a - 8b3 3)
= 2 a - 2b (a + 2ab + 4b( ) 2 2)
6 x + 4x + 16
( ) 4 2
= x + 8x + 16 - 4x4 2 2
= (x + 4 - 4x2 )2 2
= (x + 4 + 2x (x + 4 - 2x2 ) 2 )
= (x + 2x + 4 (x - 2x + 42 ) 2 ) 7 2a + 3b - 6ab - 1
( )= 6ab - 2a + 3b - 1
= 2a 3b - 1 + 3b - 1( )
= 2a + 1 3b - 1( )( )
8 5x + 8x - 4 ( ) 2
= x + 2 5x - 2( )( )
9 x - 6x + 12x - 8 = x - 2
( ) 3 2 ( )3 ( )10 x - xy - x - y - 22
= -xy - y + x - x - 22
= -y x + 1 + x - 2 x + 1( ) ( )( )
= x + 1 x - y - 2( )( ) 11 (x + y - (x - y
( ) )4 )4
= (x + y + (x - y{ )2 )2}{(x + y - (x - y)2 )2}
= (2x + 2y 4xy2 2)
= 8xy(x + y2 2)
12 4x + 1
( ) 4
= 4x + 4x + 1 - 4x4 2 2
= (2x + 1 - 4x2 )2 2
= (2x + 1) + 2x (2x + 1) - 2x2 2
= (2x + 2x + 1 (2x - 2x + 12 ) 2 ) 13 x + 1 x + 2 x + 3 x + 4 - 24
( ) ( )( )( )( )
= x + 1 x + 4 x + 2 x + 3 - 24( )( )( )( ) ( )14 abc + ab + bc + ca + a + b + c + 1
= abc + ab + ca + a + bc + b + c + 1
