phân tích đa thức thành nhân tử
a) \(x^3+y^3+z^3-3xyz\)
b) \(x^4+2011x^2+2010x+2011\)
Phân tích đa thức thành nhân tử
\(x^4+2011x^2+2010x+2011\)
=(x4−x3+2011x2)+
(x3−x2+2011x)+(x2−x+2011)
=x2(x2−x+2011)+x(x2−x+2011)+(x2−x+2011)
=(x2+x+1)(x2−x+2011)
=(x4−x3+2011x2)+(x3−x2+2011x)+(x2−x+2011)
=x2(x2−x+2011)+x(x2−x+2011)+(x2−x+2011)
=(x2+x+1)(x2−x+2011)
x3−x2+2011x)+(x2−x+2011)
=x2(x2−x+2011)+x(x2−x+2011)+(x2−x+2011)=(x2+x+1)(x2−x+2011)
phân tích đa thức thành nhân tử
a, ( x - 3)^2 - ( 5 - 2x )^2 = 0
b, ( x+ y )^2 - x + 4xy - 4y^2
c, ( x+y )^3 - ( x - y )^3
d, x^3 + y^3 + z^3 - 3xyz
\(a,\Rightarrow\left(x-3-5+2x\right)\left(x-3+5-2x\right)=0\\ \Rightarrow\left(3x-8\right)\left(2-x\right)=0\Rightarrow\left[{}\begin{matrix}x=2\\x=\dfrac{8}{3}\end{matrix}\right.\\ b,=\left(x+y\right)^2-\left(x-2y\right)^2\\ =\left(x+y-x+2y\right)\left(x+y+x-2y\right)=3y\left(2x-y\right)\\ c,=\left(x+y-x+y\right)\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\\ =2y\left(3x^2+y^2\right)\\ d,=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
Phân tích các đa thức sau thành nhân tử
a) (x+y+z)^3 - x^3 - y^3 - z^3
b) x^4 + 2012x^2 + 2011x + 2012
= x3 + y3 + z3 + 3x2yz + 3xy2z + 3xyz2 - x3 -y3 - z3
=3x2yz + 3xy2z + 3xyz2
= 3xyz( x + y + z)
b.
x^4+2012x^2+2012x-x+2012=
(x^4-x)+2012(x^2+x+1)=
x(x-1)(x^2+x+1)+2012(x^2+x+1)=
(x+2012)(x^2+x+1)
Phân tích thành nhân tử :
a ) \(3x^3-7x^2+17x-5\)
b ) \(x^4+2011x^2+2010x+2011\)
a ) \(3x^3-7x^2+17x-5\)
\(=\left(3x^2-x^2\right)-\left(6x^2-2x\right)+\left(15x-5\right)\)
\(=x^2\left(3x-1\right)-2x\left(3x-1\right)+5\left(3x-1\right)\)
\(=\left(x^2-2x+5\right)\left(3x-1\right)\)
b \(x^4+2011x^2+2010x+2011\)
\(=x^4-x+2011x^2+2011x+2011\)
\(=x\left(x^3-1\right)+2011\left(x^2+x+1\right)\)
\(=x\left(x-1\right)\left(x^2+x+1\right)+2011\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2-x+2011\right)\)
Phân tích đa thức sau thành nhân tử: x4+2011x2+2010x+2011
x4+2011x2+2010x+2011
=(x4+x3+x2)+(2011x2+2011x+2011)-(x3+x2+x)
=x2(x2+x+1)+2011(x2+x+1)-x(x2+x+1)
=(x2+x+1)(x2+2011-x)
x4+2011x2+2010x+2011=x4-x+2011x2+2011x+2011
=x(x3-1)+2011(x2+x+1)
=x(x- 1)(x2+x+1)+2011(x2+x+1)
=(x2+x+1)[x(x-1)+2011]
=(x2+x+1)(x2-x+2011)
Phân tích các đa thức sau thành nhân tử:
a) \(\left(x+y+z\right)^3-x^3-y^3-z^3\)
b) \(z^4+2010x^2+2009x+2010\)
a) (x + y + z)3 - x3 - y3 - z3
= (x + y + z)3 - z3 - (x3 + y3)
= (x + y + z - z)[(x + y + z)2 + (x + y + z).z + z2) - (x + y)(x2 - xy + y2)
= (x + y)(x2 + y2 + z2 + 2xy + 2yz + 2zx + 2xz + 2yz + z2 + z2) - (x + y)(x2 - xy + y2)
= (x + y)(x2 + y2 + 3z2 + 2xy + 4yz + 4zx) - (x + y)(x2 - xy + y2)
= (x + y)(3z2 + 3xy + 5yz + 4zx)
b) Sửa đề x4 + 2010x2 + 2009x + 2010
= (x4 + x2 + 1) + (2009x2 + 2009x + 2009)
= (x4 + 2x2 + 1 - x2) + 2009(x2 + x + 1)
= [(x2 + 1)2 - x2] + 2009(x2 + x + 1)
= (x2 + x + 1)(x2 - x + 1) + 2009(x2 + x + 1)
= (x2 + x + 1)(x2 - x + 2010)
Bài 5. Phân tích các đa thức thành nhân tử
a) (x2-4x)2-8(x2-4x)+15 b) (x2+2x)2+9x2+18x+20
c) ( x+1)(x+2)(x+3)(x+4)-24 d) (x-y+5)2-2(x-y+5)+1
Bài 6. Phân tích các đa thức thành nhân tử
a) x2y+x2-y-1 b) (x2+x)2+4(x2+x)-12
c) (6x+5)2(3x+2)(x+1)-6
Phân tích đa thức sau thành nhân tử:
a) (x+y+z)3 - x3 - y3 - z3
b) x4 + 2010x2 + 2009x + 2010
Phân tích các đa thức sau thành nhân tử
a) (x+y+z)3 - x3 - y3 -z3
b) x4 + 2012x2 + 2011x +2012
a) \(\left(x+y+z\right)^3-x^3-y^3-z^3\)
\(=\left[\left(x+y\right)+z\right]^3-x^3-y^3-z^3\)
\(=\left[\left(x+y\right)^3+z^3+3.\left(x+y\right).z.\left(x+y+z\right)\right]-x^3-y^3-z^3\)
\(=\left[x^3+y^3+3xy.\left(x+y\right)+z^3+3\left(x+y\right).z.\left(x+y+z\right)\right]-x^3-y^3-z^3\)
\(=3xy\left(x+y\right)+3\left(x+y\right)z.\left(x+y+z\right)\)
\(=3.\left(x+y\right)\left(xy+zx+zy+z^2\right)\)
\(=3.\left(x+y\right)\left(y+z\right)\left(z+x\right)\)
b) \(x^4+2012x^2+2011x+2012\)
\(=x^4-x+2012x^2+2012x+2012\)
\(=x.\left(x^3-1\right)+2012.\left(x^2+x+1\right)\)
\(=x.\left(x-1\right)\left(x^2+x+1\right)+2012.\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2-x+2012\right)\)