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a) a4 + a2 - 2
a4 + 2a2 - a2 - 2
a2.( a2 + 2 ) - ( a2 + 2 )
( a2 - 1 ).( a2 + 2 )
( a + 1 ).( a - 1 ).( a2 +2 )
b) x4 + 4x2 - 5
x4 + 5x2 - x2 - 5
x2.( x2 + 5 ) - ( x2 + 5 )
( x2 - 1 ).( x2 + 5 )
( x + 1 ).( x - 1 ).( x2 + 5 )
c) x3 - 19x - 30
x3 + 2x2 - 2x2 + 4x - 15x - 30
x2( x + 2 ) - 2x.( x + 2 ) - 15.( x + 2 )
( x + 2 ).( x2 - 2x - 15 )
d) x3 - 7x - 6
x3 - 3x2 + 3x2 - 9x + 2x - 6
x2.( x - 3 ) + 3x.( x - 3 ) + 2.( x - 3 )
( x - 3 ).( x2 + 3x +2 )
( x - 3 ).( x2 + 2x + x + 2 )
( x - 3 ).( x.( x + 2 ) + ( x + 2 )
( x + 1 ).( x + 2 ).( x - 3 )
e) x3 - 5x2 - 14x
x3 - 7x2 + 2x2 - 14x
x2.( x - 7 ) + 2x.( x - 7 )
( x - 7 ).( x2 + 2x )
x.( x + 2 ).( x - 7 )
(x - 4)(x2 + 4x + 16) - x(x2 - 6) = 2
x3 - 64 - x3 + 6x = 2
6x = 2 + 64
6x = 66
x = 66 : 6
x = 11
x3 - 27 + 3x(x - 3)
= (x - 3)(x2 + 3x + 9) + 3x(x - 3)
= (x - 3)(x2 + 3x + 9 + 3x)
= (x - 3)(x2 + 6x + 9)
= (x - 3)(x + 3)2
5x3 - 7x2 + 10x - 14
= 5x(x2 + 2) - 7(x2 + 2)
= (x2 + 2)(5x - 7)
a)x^2-(a+b)x+ab
= x^2 - ax - bx + ab
= (x^2 - ax) - (bx - ab)
= x(x-a) - b(x-a)
= (x-b)(x-a)
b)7x^3-3xyz-21x^2+9z
=
c)4x+4y-x^2(x+y)
= 4(x + y) - x^2(x+y)
= (4-x^2) (x+y)
= (2-x)(2+x)(x+y)
d) y^2+y-x^2+x
= (y^2 - x^2) + (x+y)
= (y-x)(y+x)+ (x+y)
= (y-x+1) (x+y)
e)4x^2-2x-y^2-y
= [(2x)^2 - y^2] - (2x +y)
= (2x-y)(2x+y) - (2x+y)
= (2x -y -1)(2x+y)
f)9x^2-25y^2-6x+10y
=
b ) Ta có : 3x2 - 7x - 6
= 3x2 - 9x + 2x - 6
= 3x (x - 3) + 2(x - 3)
= (x - 3)(3x + 2)
Bài 4.
a) 3xy2 - 45x2y = 3xy( y - 15x )
b) 25y2 - 4x2 + 4x - 1
= 25y2 - ( 4x2 - 4x + 1 )
= ( 5y )2 - ( 2x - 1 )2
= ( 5y - 2x + 1 )( 5y + 2x - 1 )
c) x2 - 5x + xy - 5y
= x( x - 5 ) + y( x - 5 )
= ( x - 5 )( x + y )
d) x2 - 8x - 33
= x2 + 3x - 11x - 33
= x( x + 3 ) - 11( x + 3 )
= ( x + 3 )( x - 11 )
Bài 5.
a) A = ( x - 2 )3 - x2( x - 4 ) + 8
= x3 - 6x2 + 12x - 8 - x3 + 4x2 + 8
= -2x2 + 12x
B = ( x2 - 6x + 9 ) : ( x - 3 ) - x( x + 7 ) - 9
= ( x - 3 )2 : ( x - 3 ) - x2 - 7x - 9
= x - 3 - x2 - 7x - 9
= -x2 - 6x - 12
b) Với x = -1 thì A = -2.(-1)2 + 12.(-1) = -2 - 12 = -14
a, ( x2 + x )2 - 14 ( x2 + x ) + 24
= (x2 + x)2 - 2(x2 + x) -12(x2 + x) + 24
= (x2 + x).(x2 + x -2) - 12(x2 + x -2)
= (x2 + x -2).(x2 + x -12)
= (x2 + 2x - x - 2).(x2 + 4x - 3x - 12)
=[x.(x+2)-(x+2)].[x.(x+4)-3(x+4)]
= (x+2).(x-1).(x+4).(x-3)
= x4 + 2x3 - 13x2 - 14x + 24
b, ( x2 + x )2 + 4x2 + 4x - 12
= x4 + 2x3 + x2 + 4x2 + 4x -12
= x4 + 2x3 + 5x2 + 4x -12
c, x4 + 2x3 + 5x2 + 4x - 12
= x4 - x3 + 3x3 - 3x2 + 8x2 - 8x +12x -12
= x3(x-1) + 3x2(x-1) + 8x(x-1) + 12(x-1)
= (x-1) . (x3 + 3x2 + 8x +12)
= (x-1) . ( x3 +2x2 + x2 + 2x + 6x +12)
= (x-1). [x2(x+2) + x(x+2) + 6(x+2)]
= (x-1).(x+2).(x2 + x+ 6)
a)x7+x5+1=x7+x6-x6+2x5-x5+x4-x4+x3-x3+x2-x2+1
=x7-x6+x5-x3+x2+x6-x5+x4-x2+x+x5-x4+x3-x+1
=x2(x5-x4+x3-x+1)+x(x5-x4+x3-x+1)+1(x5-x4+x3-x+1)
=(x2+x+1)(x5-x4+x3-x+1)
b)4x4-32x2+1=4x4+12x3+2x2-12x3-36x2-6x+2x2+6x+1
=2x2(2x2+6x+1)-6x(2x2+6x+1)+1(2x2+6x+1)
=(2x2-6x+1)(2x2+6x+1)
c)x6+27=(x2+3)(x2-3x+3)(x2+3x+3)
d)3(x4+x2+1)-(x2+x+1)
=3x4-3x3+2x2+3x3-3x2+2x+3x2-3x+2
=x2(3x2-3x+2)+x(3x2-3x+2)+1(3x2-3x+2)
=(x2+x+1)(3x2-3x+2)
e)bạn tự làm nhé
\(x^8+x^7+1\)
\(=x^8+x^7+x^6-x^6+x^5-x^5+x^4-x^4+x^3-x^3+x^2-x^2+x-xx+1\)
\(=\left(x^8-x^6+x^5-x^3+x^2\right)\)
\(+\left(x^7-x^5+x^4-x^2+x\right)\)
\(+\left(x^6-x^4+x^3-x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)
Bài 2 : Phân tích các đa thức sau thành nhân tử :
a, x2 + 7x + 12
= x2 + 3x + 4x + 12
= x(x + 3) + 4(x + 3)
= (x + 3)(x + 4)
b, 3x2 - 8x + 5
= 3x2 - 3x - 5x + 5
= 3x(x - 1) - 5(x - 1)
= (x - 1)(3x - 5)
c, x4 + 5x2 - 6
= x4 - x2 + 6x2 - 6
= x2(x2 - 1) + 6(x2 - 1)
= (x2 - 1)(x2 + 6)
= (x - 1)(x + 1)(x2 + 6)
d, x4 - 34x2 + 225
= x4 - 9x2 - 25x2 + 225
= x2(x2 - 9) - 25(x2 - 9)
= (x2 - 9)(x2 - 25)
= (x - 3)(x + 3)(x - 5)(x + 5)
e, x2 - 5xy + 6y2
= x2 + xy - 6xy + 6y2
= x(x + y) - 6y(x + y)
= (x + y)(x - 6y)
f, 4x2 - 17xy + 13y2
= 4x2 - 4xy - 13xy + 13y2
= 4x(x - y) - 13y(x - y)
= (x - y)(4x - 13y)
Bài 4:
a) Ta có: \(a^4+a^2+1\)
\(=a^4+2a^2+1-a^2\)
\(=\left(a^2+1\right)^2-a^2\)
\(=\left(a^2-a+1\right)\left(a^2+a+1\right)\)
b) Ta có: \(a^4+a^2-2\)
\(=a^4+2a^2-a^2-2\)
\(=a^2\left(a^2+2\right)-\left(a^2+2\right)\)
\(=\left(a^2+2\right)\left(a^2-1\right)\)
\(=\left(a^2+2\right)\left(a-1\right)\left(a+1\right)\)
c) Ta có: \(x^4+4x^2-5\)
\(=x^4+5x^2-x^2-5\)
\(=x^2\left(x^2+5\right)-\left(x^2+5\right)\)
\(=\left(x^2+5\right)\left(x^2-1\right)\)
\(=\left(x^2+5\right)\left(x-1\right)\left(x+1\right)\)
d) Ta có: \(x^3-19x-30\)
\(=x^3-25x+6x-30\)
\(=x\left(x^2-25\right)+6\left(x-5\right)\)
\(=x\left(x-5\right)\left(x+5\right)+6\left(x-5\right)\)
\(=\left(x-5\right)\left(x^2+5x\right)+6\left(x-5\right)\)
\(=\left(x-5\right)\left(x^2+5x+6\right)\)
\(=\left(x-5\right)\left(x^2+2x+3x+6\right)\)
\(=\left(x-5\right)\left[x\left(x+2\right)+3\left(x+2\right)\right]\)
\(=\left(x-5\right)\left(x+2\right)\left(x+3\right)\)
e) Ta có: \(x^3-7x-6\)
\(=x^3-4x-3x-6\)
\(=x\left(x^2-4\right)-3\left(x+2\right)\)
\(=x\left(x-2\right)\left(x+2\right)-3\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x\right)-3\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x-3\right)\)
\(=\left(x+2\right)\left(x^2-3x+x-3\right)\)
\(=\left(x+2\right)\left[x\left(x-3\right)+\left(x-3\right)\right]\)
\(=\left(x+2\right)\left(x-3\right)\left(x+1\right)\)
f) Ta có: \(x^3-5x^2-14x\)
\(=x\left(x^2-5x-14\right)\)
\(=x\left(x^2-7x+2x-14\right)\)
\(=x\left[x\left(x-7\right)+2\left(x-7\right)\right]\)
\(=x\left(x-7\right)\left(x+2\right)\)