1.103
a) (x^2-8)^2 + 36=x^4 -16x^2 + 64+36
                           =(x^2)^2 -(4x)^2 +100
                           =(x^2 -4x)(x^2+4x)+100
b) x^8 + x^4 + 1=(x^4)^2 +2x^4 +1 -x^4
                         =(x^4 + 1)^2-x^4
                         =(x^4 +1 -x^2)(x^4+1+x^2)

1.102
a) 20x^2 + 7x - 6
<=> 20x^2 + 15x - 8x - 6
<=> 5x(4x + 3) - 2(4x + 3)
<=> (5x - 2)(4x + 3)
b) 18x^2 + 21x - 4
<=> 18x^2 + 24x - 3x - 4
<=> 6x(3x + 4) - (3x + 4)
<=> (6x - 1)(3x + 4)
c) 12x^2 - 23xy +10y^2
<=> 12x^2 - 15xy - 8xy + 10y^2
<=> 3x(4x - 5y) - 2y(4x - 5y)
<=> (3x - 2y)(4x - 5y)

Bài 102
a) 20x^2 + 7x - 6
= ( 4x + 3 ).( 5x - 2 )
b) 18x^2 + 21x - 4
= ( 3x + 4 ).( 6x - 1 )
c) 12x^2 - 23xy + 10y^2
= ( 2y - 3x ).( 5y - 4x )
d) x^4 - 5x^2y^2 + 4y^4
= ( x^2 - y^2 ).( x^2 - 4y^2 )
= ( x - y ).( x + y ).( x - 2y ).( x + 2y )
Bài 105
a) 6x^3 - 11x^2 - x - 2
= ( x - 2 ).( 6x^2 + x + 1 )
b) 4x^3 + 5x^2 + 5x + 1
= ( 4x + 1 ).( x^2 + x + 1 )
 

Bài 102
a) 20x^2 + 7x - 6
= ( 4x + 3 ).( 5x - 2 )
b) 18x^2 + 21x - 4
= ( 3x + 4 ).( 6x - 1 )
c) 12x^2 - 23xy + 10y^2
= ( 2y - 3x ).( 5y - 4x )
d) x^4 - 5x^2y^2 + 4y^4
= ( x^2 - y^2 ).( x^2 - 4y^2 )
= ( x - y ).( x + y ).( x - 2y ).( x + 2y )
Bài 105
a) 6x^3 - 11x^2 - x - 2
= ( x - 2 ).( 6x^2 + x + 1 )
b) 4x^3 + 5x^2 + 5x + 1
= ( 4x + 1 ).( x^2 + x + 1 )
 

Bài 1.61
a) ( x - 9 ).( x - 7 ) + 1
Đặt x - 8 = a ta được :
( a - 1 ).( a + 1 ) + 1
= a^2 - 1 + 1
= a^2
= ( x - 8 )^2
b) x^6 - y^6
= ( x^2 - y^2 ).( x^4 + x^2y^2 + y^4) \
= ( x - y ).( x + y ).( x^4 + x^2y^2 + y^4)
Bài 1.62
a) ( x + 2y - 3 )^2 - 4.( x + 2y - 3 ) + 4
Đặt ( x + 2y - 3 ) = a
=> a^2 - 4a + 4
= ( a - 2 )^2
= ( x + 2y - 3 - 2 )^2
= ( x + 2y - 5 )^2

1.77
a) 5x^2(x - 1) +10xy(x - 1) - 5y^2(1 - x)
<=> (5x^2 +10xy + 5y^2)(x - 1)
<=> 5(x^2 +2xy + y^3)(x - 1)
<=> 5(x + y)^2.(x - 1)
b) x^5 - x^4y - xy^4 + y^5
<=> (x^5 - x^4y) - (xy^4 - y^5)
<=> x^4(x - y) - y^4(x - y)
<=> (x^4 - y^4)(x - y)
<=> (x^2 - y^2)(x - y)(x^2 + y^2)
<=> (x -y)(x - y)(x +y)(x^2 +y^2)
<=> (x +y)(x^2 +y)(x - y)^2

1.79
a) x^3(x^2 +1)^2 - 49x
<=> x(x^2(x^2 +1)^2 - 49)
<=> x{[x(x^2 +1)]^2 -7^2}
<=> x(x^3 +x - 7)(x^3 +x +7)
b) (x^2- 9)^2 + 12x(x- 3)^2
<=> (x - 3)^2.(x + 3)^2 + 12x(x - 3)^2
<=> (x - 3)^2.[(x+ 3)^2 +12x]
<=> (x - 3)^2.(x^2 + 6x +9 +12x)
<=> (x^2 + 18x + 9)(x- 3)^2
c) (x - z)(x + z) - y(2x - y)
<=> x^2 - z^2 - 2xy +y^2
<=> (x^2 - 2xy +y^2) - z^2
<=> (x - y)^2 - z^2
<=> (x - y - z)(x - y +z)

1.80
a) x^2 - 6xy + 9y^2 - 9
<=> (x^2 - 6xy +9y^2) - 9
<=> (x - 3y)^2-9
<=> (x - 3y - 3)(x - 3y + 3)
b) 25x^2 - y^2 +4y - 4
<=> 25x^2 - (y^2 - 4y +4)
<=> 25x^2 - (y - 2)^2
<=> (5x - y + 2)(5x +y - 2)
c) a^2 + b^2 - x^2 - y^2 + 2ab - 2xy
<=> (a^2 + 2ab +b^2) -(x^2 +2xy + y^2)
<=> (a + b)^2 - (x + y)^2
<=> (a + b - x - y)(a + b +x + y)

1.73
Gọi A = x^3 + x^2.z +  y^2.z - xyz + y^3
         = (x^3 + y^3) + z(x^2 - xy + y^2)
         = (x+y)(x^2 - xy + y^2) + z(x^2 - xy + y^2)
         = (x^2 - xy + y^2)(x+y+z)
mà x+y+z = 0
=> A = (x^2 - xy + y^2).0
        = 0
Chứng tỏ với x+y+z = 0 thì x^3 + x^2.z +  y^2.z - xyz + y^3 = 0

1.72
A = 2^10 - 2^9 - 2^8 - 2^7 -...- 2^2 - 2 - 1
=> 2A = 2^11 - 2^10 - 2^9 - 2^8 -...- 2^3 - 2^2 - 2
=> 2A - A = 2^11 - 2^10 - 2^9 - 2^8 -...- 2^3 - 2^2 - 2 - (2^10 - 2^9 - 2^8 - 2^7 -...- 2^2 - 2 - 1)
=> A = 2^11 - 2^10 - 2^10 + 1
=> A = 2^11 - 2^11 + 1
=> A = 1
1.69
abc + ab + bc + ac + a + b + c + 1
= (abc+ab) + (bc+b) + (ac+a) + (c+1)
= ab(c+1) + b(c+1) + a(c+1) + (c+1)
= (c+1)(ab+b+a)

1.67
a, x^3 + 2x^2 - 3x - 6
= x^2.(x+2) - 3(x+2)
= (x+2)(x^2 - 3)
b, 2a^2.c^2 - 2abc + bd - acd
= 2ac(ac-b) - d(ac-b)
= (ac-b)(2ac-d)
c, 12x^2 - 3xy - 8xz + 2yz
= 3x(4x-y) - 2z(4x-y)
= (4x-4)(3x-2z)

1.64
a, x^3 + 15x^2 + 75x + 125 = 0
<=> x^2 + 5x^2 + 10x^2 + 50x + 25x + 125 = 0
<=> (x+5)(x^2 + 10x + 25) = 0
<=> (x+5)(x+5)^2 = 0
<=> (x+5)^3 = 0
<=> x+5 = 0
<=> x = -5
b, (x+2)^3 + (x-3)^3 = 0
<=> (x+2+x-3)[(x+2)^2 - (x+2)(x-3) + (x-3)^2] = 0
<=> (2x-1)(x^2 + 4x + 4 - x^2 + x + 6 + x^2 - 6x + 9) = 0
<=> (2x-1)(x^2 - x + 19) = 0
<=> (2x-1)[(x-1/2)^2 + 75/4] = 0
<=> 2x-1 = 0 (Vì (x-1/2)^2 + 75/4 > 0)
<=> x = 1/2

1.63
(x^2 + y^2 - 17)^2 - 4(xy-4)^2
= (x^2 + y^2 - 17)^2 - [2(xy-4)]^2
= (x^2 + y^2 - 17)^2 - (2xy-8)^2
= (x^2 + y^2 - 17 + 2xy - 8)(x^2 + y^2 - 17 - 2xy + 8)
= (x^2 + 2xy + y^2 - 25)(x^2 - 2xy + y^2 - 9)
= [(x+y)^2 - 5^2][(x-y)^2 - 3^2]
= (x+y-5)(x+y+5)(x-y-3)(x-y+3)

1.62
a, (x+2y-3)^2 - 4(x+2y-3) + 4
= (x+2y-3)^2 - 2(x+2y-3) - 2(x+2y-3) + 4
= (x+2y-3)(x+2y-3-2) - 2(x+2y-3-2)
= (x+2y-3-2)(x+2y-3-2)
= (x+2y-5)((x+2y-5)
= (x+2y-5)^2
b, (x-y)^3 - 1 - 3(x-y)(x-y-1)
= (x-y-1)[(x-y)^2 + x-y + 1) - (3x-3y)(x-y-1)
= (x-y-1)(x^2 - 2xy + y^2 + 1 + x - y - 3x + 3y)
= (x-y-1)(x^2 - 2xy + y^2 + 1 - 2x + 2y)

1.61
a, (x-9)(x-7) + 1
= x^2 - 16x + 63 + 1
= x^2 - 16x + 64
= (x-8)^2
b, x^6 - y^6
= (x^3 - y^3)(x^3 + y^3)
= (x-y)(x^2 + xy + y^2)(x+y)(x^2 - xy + y^2)
c, (x^2 + x - 1)^2 + 4x^2 + 4x
= x^4 + x^2 + 1 + 2x^3 - 2x^2 - 2x + 4x^2 + 4x
= x^4 + 2x^3 + 3x^2 + 2x + 1
= x^4 + x^3 + x^2 + x^3 + x^2 + x + x^2 + x + 1
= x^2.(x^2 + x + 1) + x(x^2 + x + 1) + (x^2 + x + 1)
= (x^2 + x + 1)(x^2 + x + 1)
= (x^2 + x + 1)^2

1.106
b, B = x^4 - 5x^3 + 10x + 4
       = x^4 + x^3 - 6x^3 - 6x^2 + 6x^2 + 6x + 4x + 4
       = (x+1)(x^3 - 6x^2 + 6x + 4)
       = (x+1)(x^3 - 2x^2 - 4x^2 + 8x - 2x + 4)
       = (x+1)(x-2)(x^2 - 4x - 2)
c, C = x^4 + 3x^3 + 3x^2 + 3x + 2
       = x^4 + x^3 + 2x^3 + 2x^2 + x^2 + x + 2x + 2
       = (x+1)(x^3 + 2x^2 + x + 2)
       = (x+1)(x+2)(x^2 + 1)

1.107
a, 4x^2 - 12x - y^2 + 8y - 7
= 4x^2 - 12x + 9 - y^2 + 8y - 16
= (2x-3)^2 - (y-4)^2
= (2x-3-y+4)(2x-3+y-4)
= (2x-y+1)(2x+y-7)
b, 2x^4 - x^3.y + 3x^2.y^2 - xy^3 + 2y^4
= (2x^4 + 2x^2.y^2 + 2y^4) - (x^3.y + x^2.y^2 - xy^3)
= 2(x^4 + 2x^2.y^2 + y^4 - x^2.y^2) - xy(x^2 - xy + y^2)
= 2[(x^2 + y^2)^2 - x^2.y^2] - xy(x^2 - xy + y^2)
= 2(x^2 + y^2 - xy)(x^2 + y^2 + xy) - xy(x^2 - xy + y^2)
= (x^2 + y^2 - xy)(2x^2 + 2y^2 + 2xy - xy)
= (x^2 + y^2 - xy)(2x^2 + 2y^2 + xy)