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1, 2x2 - 8xy - 5x + 20y
= (2x2 - 5x) - (8xy - 20y)
= x(2x - 5) - 4y(2x - 5)
= (2x - 5) (x - 4y)
2, x3 - x2y - xy + y2
= (x3 - xy) - (x2y - y2)
= x(x2 - y) - y(x2 - y)
= (x2 - y) (x - y)
3, x2 - 2xy - 4z2 + y2
= (x2 - 2xy + y2) - 4z2
= (x - y)2 - (2z)2
= (x - y - 2z) (x - y + 2z)
4, a3 + a2b - a2c - abc
= (a3 - a2c) + (a2b - abc)
= a2(a - c) + ab(a - c)
= (a - c) (a2 + ab)
5, x3 + y3 + 3x2y + 3xy2 - x - y
= (x3 + 3x2y + 3xy2 + y3) - (x + y)
= (x + y) 3 - (x + y)
= (x + y) [(x + y)2 - 1]
= (x + y) (x + y - 1) (x + y + 1)
1.
= 4x\(^{^{ }2}\)-4x-9x+9
=4x(x-1)-9(x-1)
=(4x-9)(x-1)
1) 2x2-8xy-5x+20y
=2x(x-4y)-5(x-4y)
=(2x-5)(x-4y)
2) x3-x2y-xy+y2
=x2(x-y)-y(x-y)
=(x2-y)(x-y)
3) x2-2xy-4z2+y2
=(x-y)2-(2z)2
=(x-y-2z)(x-y+2z)
4) a3+a2b-a2c-abc
=a2(a+b)-ac(a+b)
=(a2-ac)(a+b)
=a(a-c)(a+b)
5) x3+y3+3x2y+3xy2-x-y
=(x+y)(x2-xy+y2)+3xy(x+y)-(x+y)
=(x+y)(x2-xy+y2+3xy-1)
=(x+y)[(x+y)2-1)]
=(x+y)(x+y+1)(x+y-1)
6) x3+x2y-x2z-xyz
=x2(x+y)-xz(x+y)
=(x2-xz)(x+y)
=x(x-z)(x+y)
7) =[x(y+z)2-2xyz]+[y(z+x)2-2xyz]+z(x+y)2
=x(y2+z2)+y(z2+x2)+z(x+y)2
=xy(x+y)+z2(x+y)+z(x+y)2
=(x+y)(xy+z2+zx+zy)
=(x+y)(x+z)(y+z)
8) x3(z-y)+y3(x-z)+z3(y-x)
Tách x-z= -[z-y+y-x]
Tạm thời phân tích như sau:
i) x4 - 2x3 + 2x - 1
= (x4 - 1) - (2x3 - 2x)
= (x2 + 1).(x2 -1) - 2x.(x2 - 1)
= (x2 - 1).(x2 - 2x + 1)
j) a6 - a4 + 2a3 + 2a2
= (a3 + a2).(a3 - a2) + 2.(a3 + a2)
= (a3 + a2).(a3 - a2 +2)
k) x4 - x3 + 2x2 + x + 1 (tạm thời giải thế này)
= x3.(x - 1) + (2x + 3 - \(\frac{4}{x-1}\)).(x -1)
= (x - 1).(x3 + 2x + 3 - \(\frac{4}{x-1}\))
Nếu đề là:
x4 + x3 + 2x2 + x + 1
= x4 + x2 + x3 + x + x2 + 1
= x2.(x2 + 1) + x.(x2 + 1) + x2 + 1
= (x2 + 1).(x2 + x + 1)
m) x2y + xy2 + x2z + y2z + 2xyz
= xy.(x + y) + z.(x2 + 2xy + y2)
= xy.(x + y) + z.(x + y).(x + y)
= (x + y).(xy + xz + yz)
n) x5 + x4 + x3 + x2 + x + 1
= x4.(x + 1) + x2.(x + 1) + x + 1
= (x + 1).(x4 + x2 + 1)
\(x^4-2x^3+2x-1\)
\(=\left(x^4-1\right)-2x\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left(x^2+1\right)-2x\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left(x^2-2x+1\right)\)
\(=\left(x-1\right)\left(x+1\right)\left(x-1\right)^2\)
\(=\left(x-1\right)^3\left(x+1\right)\)
a) 16x2 - 9
= ( 4x )2 - 32
= ( 4x - 3 )( 4x + 3 )
b) 9a2 - 25b4
= ( 3a )2 - ( 5b2 )2
= ( 3a - 5b2 )( 3a + 5b2 )
c) 81 - y4
= 92 - ( y2 )2
= ( 9 - y2 )( 9 + y2 )
= ( 32 - y2 )( 9 + y2 )
= ( 3 - y )( 3 + y )( 9 + y2 )
d) ( 2x + y )2 - 1
= ( 2x + y )2 - 12
= ( 2x + y - 1 )( 2x + y + 1 )
e) ( x + y + z )2 - ( x - y - z )2
= [ x + y + z - ( x - y - z ) ][ x + y + z + ( x - y - z ) ]
= [ x + y + z - x + y + z ][ x + y + z + x - y - z ]
= [ 2y + 2z ].2x
= 2[ y + z ].2x
= 4x[ y + z ]
a) ( 2x + 3 )2 - 2( 2x + 3 )( 2x + 5 ) + ( 2x + 5 )2
= [ ( 2x + 3 ) - ( 2x + 5 ) ]2
= ( 2x + 3 - 2x - 5 )2
= (-2)2 = 4
b) ( x2 + x + 1 )( x2 - x + 1 )( x2 - 1 )
= ( x4 - x3 + x2 + x3 - x2 + x + x2 - x + 1 )( x2 - 1 )
= ( x4 + x2 + 1 )( x2 - 1 )
= x6 - x4 + x4 - x2 + x2 - 1
= x6 - 1
c) ( x + y )2 + ( x - y )2
= x2 + 2xy + y2 + x2 - 2xy + y2
= 2x2 + 2y2 = 2( x2 + y2 )
d) 2( x - y )( x + y ) + ( x + y )2 + ( x - y )2
= [ ( x + y ) + ( x - y ) ]2
= ( x + y + x - y )2
= ( 2x )2 = 4x2
e) ( x - y + z )2 + ( z - y )2 + 2( x - y + z )( y - z )
= ( x - y + z )2 + ( z - y )2 - 2( x - y + z )( z - y )
= [ ( x - y + z ) - ( z - y ) ]2
= ( x - y + z - z + y )2
= x2
f) ( a + b - c )2 + ( a - b + c )2 - 2( b - c )2
= [ ( a + b ) - c ]2 + [ ( a - b ) + c ]2 - 2( b2 - 2bc + c2 )
= [ ( a + b )2 - 2( a + b )c + c2 ] + [ ( a - b )2 + 2( a - b )c + c2 ] - 2b2 + 4bc - 2c2
= a2 + b2 + c2 + 2ab - 2bc - 2ca + c2 + a2 + b2 + c2 - 2ab + 2bc + 2ac - 2b2 + 4bc - 2c2
= 2a2
g) ( a + b + c )2 + ( a - b - c )2 + ( b - c - a )2 + ( c - a - b )2
= [ ( a + b ) + c ]2 + [ ( a - b ) - c ]2 + [ ( b - c ) - a ]2 + [ ( c - a ) - b ]2
= [ ( a + b )2 + 2( a + b )c + c2 ] + [ ( a - b )2 - 2( a - b )c + c2 ] + [ ( b - c )2 - 2( b - c )a + a2 ] + [ ( c - a )2 - 2( c - a )b + b2 ]
= [ a2 + b2 + c2 + 2ab + 2bc + 2ca ] + [ a2 + b2 + c2 - 2ab + 2bc - 2ca ] + [ a2 + b2 + c2 - 2ab - 2bc + 2ca ] + [ a2 + b2 + c2 + 2ab - 2bc - 2ca ]
= 4a2 + 4b2 + 4c2
Có vẻ hơi dài dòng nhỉ :( Nhưng như này là kĩ nhất đấy :)