Phân tích đa thức thành nhân tử:
a) 3x4+5x3+14x2+3x+5
b) 3x4+7x3+8x2+9x+15
Phân tích các đa thức sau thành nhân tử:
a)6x3y2.(2-x)+9x2y2.(x-2)
b)5x3+20x2+20xy-5xy2
c)8x2-10x-3
\(a,=\left(x-2\right)\left(9x^2y^2-6x^3y^2\right)=3x^2y^2\left(3-2x\right)\left(x-2\right)\\ b,=5x\left(x^2-y^2\right)+20x\left(x+y\right)=5x\left(x-y\right)\left(x+y\right)+20x\left(x+y\right)\\ =5\left(x+y\right)\left(x^2-xy+4x\right)\\ c,=8x^2+2x-12x-3=2x\left(4x+1\right)-3\left(4x+1\right)=\left(2x-3\right)\left(4x+1\right)\)
Phân tích các đa thức sau thành nhân tử:
a/ y2 - 2y b/ 3x4 – 6x3 + 3x2
c/ 27x2 ( y – 1) – 9x3 ( 1 - y) d/y3 – 2y2 + y
e/ x3 + 6x2 + 9x f/ x3 – 2x2y + xy2
g/ x( 2- x) – x + 2 h/ 3x ( x – 1) + 6( 1 – x)
\(a,=y\left(y-2\right)\\ b,=3x\left(x^2-2x+1\right)=3x\left(x-1\right)^2\\ c,=\left(y-1\right)\left(27x^2+9x^3\right)=9x^2\left(x+3\right)\left(y-1\right)\\ d,=y\left(y^2-2y+1\right)=y\left(y-1\right)^2\\ e,=x\left(x^2+6x+9\right)=x\left(x+3\right)^2\\ f,=x\left(x^2-2xy+y^2\right)=x\left(x-y\right)^2\\ g,=\left(2-x\right)\left(x+1\right)\\ h,=\left(x-1\right)\left(3x-6\right)=3\left(x-1\right)\left(x-2\right)\)
a: =y(y-2)
b: \(=3x^2\left(x^2-2x+1\right)=3x^2\left(x-1\right)^2\)
d: \(=y\left(y^2-2y+1\right)=y\left(y-1\right)^2\)
Phân tích các đa thức sau thành nhân tử:
a) x8 + x4 + 1
b) x12 - 3x6 - 1
c) 3x4 + 10x2 - 25
d) x2 - 5y2 - y4 + 2xy - 9
Lời giải:
a.
$x^8+x^4+1=(x^4)^2+2x^4+1-x^4$
$=(x^4+1)^2-(x^2)^2=(x^4+1-x^2)(x^4+1+x^2)$
$=(x^4+1-x^2)[(x^2+1)^2-x^2]$
$=(x^4-x^2+1)(x^2+1-x)(x^2+1+x)$
b.
$x^{12}-3x^6-1=(x^6-\frac{3}{2})^2-\frac{13}{4}$
$=(x^6-\frac{3}{2}-\frac{\sqrt{13}}{2})(x^6-\frac{3}{2}+\frac{\sqrt{13}}{2})$
c.
$3x^4+10x^2-25=(3x^4+15x^2)-(5x^2+25)$
$=3x^2(x^2+5)-5(x^2+5)=(x^2+5)(3x^2-5)$
$=(x^2+5)(\sqrt{3}x-\sqrt{5})(\sqrt{3}x+\sqrt{5})$
c.
$x^2-5y^2-y^4+2xy-9$
$=(x^2+2xy+y^2)-(y^4+6y^2+9)$
$=(x+y)^2-(y^2+3)^2$
$=(x+y+y^2+3)(x+y-y^2-3)$
\(a,x^8+x^4+1\\ =\left(x^8+2x^4+1\right)-x^4\\ =\left(x^4+1\right)^2-x^4\\ =\left(x^4-x^2+1\right)\left(x^4+x^2+1\right)\\ b,x^{12}-3x^6-1\\ =\left(x^{12}-2x^6+1\right)-x^6-2\\ =\left(x^6-1\right)^2-x^6-2\\ =\left(x^6-x^3-1\right)\left(x^6+x^3-1\right)-2???\\ c,3x^4+10x^2-25\\ =4x^4-\left(x^4-10x^2+25\right)\\ =4x^4-\left(x^2-5\right)^2\\ =\left(2x^2-x^2+5\right)\left(2x^2+x^2-5\right)\\ =\left(x^2+5\right)\left(3x^2-5\right)\\ d,x^2-5y^2-y^4+2xy-9\\ =\left(x^2+2xy+y^2\right)-\left(y^4+6y^2+9\right)\\ =\left(x+y\right)^2-\left(y^2+3\right)^2\\ =\left(x+y+y^2+3\right)\left(x+y-y^2-3\right)\)
a) x8+x4+1 = (x4+1)2-x4 = (x4-x2+1)(x4+x2+1)
b) x12-3x6-1 = (x6-1)2-x6 = (x6-x3-1)(x6+x3-1)
c) 3x4+10x2-25 = 4x4-(x4-10x2+25) = 4x4- (x2-5)2 = (x2+5)(3x2-5)
d) x2-5y2-y4+2xy-9 = (x+y)2-(y2+3)2 = (x+y-y2-3)(x+y+y2+3)
Phân tích đa thức thành nhân tử:
a) 2y2-3y-5
b) x2-9x-10
c) x3+y3+z3-3xyz
\(a,=\left(2x-5\right)\left(x+1\right)\\ b,=\left(x-10\right)\left(x+1\right)\\ c,=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
Bài 4. Phân tích các đa thức sau thành nhân tử:
a) 36x3 y3 - 42x2 y2
b) 3x4 y2 + 15x2 y -18xy
c) ax - bx + ab - x2
d) 3(2x -1) - 4x2 + 4x -1
a) \(=6x^2y^2\left(6xy-7\right)\)
b) \(=3xy\left(x^3y+5x-6\right)\)
c) \(=\left(ax+ab\right)-\left(bx+x^2\right)=a\left(b+x\right)-x\left(b+x\right)=\left(a-x\right)\left(b+x\right)\)
d) \(=3\left(2x-1\right)-\left(2x-1\right)^2=\left(2x-1\right)\left(3-2x+1\right)=\left(2x-1\right)\left(4-2x\right)=2\left(2x-1\right)\left(2-x\right)\)
\(a,=6x^2y^2\left(6xy-7\right)\\ b,=3xy\left(x^3y+5x-6\right)\\ c,=x\left(a-x\right)-b\left(a-x\right)=\left(x-b\right)\left(a-x\right)\\ d,=3\left(2x-1\right)-\left(2x-1\right)^2=\left(2x-1\right)\left(3-2x+1\right)=2\left(2-x\right)\left(2x-1\right)\)
Phân tích đa thức thành nhân tử:
a) x 4 - 7 x 3 + 14 x 2 - 7x + 1;
b) 3 x 2 - 22xy - 4x + 8y + 7 y 2 +1.
a) ( x 2 – 4x + 1)( x 2 – 2x + 3). b) (3x – y – 1)(x – 7y – 1).
1. Phân tích thành nhân tử
a) x2 + 7x + 10; b) x2 – 21x + 110; c) 3x2 + 12x + 9; d) 2ax2 - 16ax + 30a.
2. Phân tích thành nhân tử
a) x2 + x – 6; b) x2 – 2x – 15; c) 4x2 - 12x - 160; d) 5x2y - 10xy - 15y.
3. Phân tích thành nhân tử
a) x2 – xy – 20y2 ; b) 3x4 + 6x2y2 – 45y4 ; c) 2bx2 – 4bxy - 70y2
4. Giải phương trình
a) x2 + x = 72; b) 3x2 – 6x = 24 c) 5x3 – 10x2 = 120x.
5. Phân tích thành nhân tử
a) 3x2 -11x + 6; b) 8x2 + 10x – 3 ; c) 8x2 -2x -1 .
Phân tích đa thức thành nhân tử:
a.10x2y – 20xy2 b. x2 – y2 + 10y – 25 c. x2 – y2 + 3x – 3y
d. x3 + 3x2 – 16x – 48 e. 9x3 + 6x2 + x f. x4 + 5x3 + 15x – 9
\(a,10x^2y-20xy^2=10xy\left(x-2y\right)\\ b,x^2-y^2+10y-25=x^2-\left(y^2-10y+25\right)=x^2-\left(y-5\right)^2=\left(x-y+5\right)\left(x+y-5\right)\\ c,x^2-y^2+3x-3y=\left(x-y\right)\left(x+y\right)+3\left(x-y\right)=\left(x-y\right)\left(x+y+3\right)\\ d,x^3+3x^2-16x-48=\left(x^3+3x^2\right)-\left(16x+48\right)=x^2\left(x+3\right)-16\left(x+3\right)=\left(x+3\right)\left(x^2-16\right)=\left(x+3\right)\left(x+4\right)\left(x-4\right)\)
\(e,9x^3+6x^2+x=x\left(9x^2+6x+1\right)=x\left(3x+1\right)^2\\ f,x^4+5x^3+15x-9=\left(x^4+5x^3-3x^2\right)+\left(3x^2+15x-9\right)=x^2\left(x^2+5x-3\right)+3\left(x^2+5x-3\right)=\left(x^2+3\right)\left(x^2+5x-3\right)\)
Phân tích đa thức thành nhân tử:
a)x3-8x2+16x
b)x2+4y2+2x-4y-4xy-24
c)x4+x3-x2-2x-2
`a)x^3-8x^2+16x`
`=x(x^2-8x+16)`
`=x(x-4)^2`
`b)x^2+4y^2+2x-4y-4xy-24`
`=(x-2y)^2+2(x-2y)-24`
`=(x-2y)^2-4(x-2y)+6(x-2y)-24`
`=(x-2y-4)(x-2y+6)`
`c)x^4+x^3-x^2-2x-2`
`=x^4-2x^2+x^3-2x+x^2-2`
`=x^2(x^2-2)+x(x^2-2)+x^2-2`
`=(x^2-2)(x^2+x+1)`
cho hai đa thức M(x)=-5x3+3x4+7-9x
N(x)=-2x4+3x-5x3-7
tính M(x)=N(x)
\(M\left(x\right)+N\left(x\right)=-5x^3+3x^4+7-9x-2x^4+3x-5x^3-7\)
\(M\left(x\right)+N\left(x\right)=x^4-10x^3-12x\)