44. Phân tích các đa thức sau thành nhân tử:
a) x3 + ; b) (a + b)3 – (a – b)3
c) (a + b)3 + (a – b)3 ; d) 8x3 + 12x2y + 6xy2 + y3
e) - x3 + 9x2 – 27x + 27.
1A. Phân tích các đa thức sau thành nhân tử:
a) x3+2x; b) 3x - 6y;
c) 5(x + 3y)- 15x(x + 3y); d) 3(x-y)- 5x(y-x).
1B. Phân tích các đa thức sau thành nhân tử:
a) 4x2 - 6x; b) x3y - 2x2y2 + 5xy;
c) 2x2(x +1) + 4x(x +1); d) 2 x(y - 1) - 2
y(1 - y).
5 5
2A. Phân tích các đa thức sau thành nhân tử: a) 2(x -1)3 - 5(x -1)2 - (x - 1);
b) x(y - x)3 - y(x - y)2 + xy(x - y);
c) xy(x + y)- 2x - 2y;
d) x(x + y)2 - y(x + y)2 + y2 (x - y).
2B. Phân tích đa thức thành nhân tử: a) 4(2-x)2 + xy - 2y;
b) x(x- y)3 - y(y - x)2 - y2(x - y);
c) x2y-xy2 - 3x + 3y;
d) x(x + y)2 - y(x + y) 2 + xy - x 2 .
1A:
a: \(x^3+2x=x\left(x^2+2\right)\)
b: \(3x-6y=3\left(x-2y\right)\)
c: \(5\left(x+3y\right)-15x\left(x+3y\right)\)
\(=5\left(x+3y\right)\left(1-3x\right)\)
d: \(3\left(x-y\right)-5x\left(y-x\right)\)
\(=3\left(x-y\right)+5x\left(x-y\right)\)
\(=\left(x-y\right)\left(5x+3\right)\)
1A. a. x(x2+2)
b. 3(x-2y)
c. 5(x+3y)(1-3x)
d. (x-y) (3-5x)
1B. a. 2x(2x-3)
b.xy(x2-2xy+5)
c. 2x(x+1)(x+2)
d. 2x(y-1)+2y(y-1)=2(y-1)(x-y)
1B:
a: \(4x^2-6x=2x\left(2x-3\right)\)
b: \(x^3y-2x^2y^2+5xy\)
\(=xy\left(x^2-2xy+5\right)\)
Câu II (2,0 điểm) Phân tích các đa thức sau thành nhân tử:
a) x2 – 3x + xy – 3y
b) x3 + 10x2 + 25x – xy2
c) x3 + 2 + 3(x3 – 2)
a) Ta có: \(x^2-3x+xy-3y\)
\(=x\left(x-3\right)+y\left(x-3\right)\)
\(=\left(x-3\right)\left(x+y\right)\)
b) Ta có: \(x^3+10x^2+25x-xy^2\)
\(=x\left(x^2+10x+25-y^2\right)\)
\(=x\left(x+5-y\right)\left(x+5+y\right)\)
c) Ta có: \(x^3+2+3\left(x^3-2\right)\)
\(=4x^3-4\)
\(=4\left(x-1\right)\left(x^2+x+1\right)\)
Phân tích các đa thức sau thành nhân tử:
a) 3x2 + xy - 4y2
b) x8 - 5x4 + 4
c) x3 + 3x2 + 3x - 7
Lời giải:
a.
$3x^2+xy-4y^2=(3x^2-3xy)+(4xy-4y^2)=3x(x-y)+4y(x-y)=(x-y)(3x+4y)$
b.
$x^8-5x^4+4=(x^8-x^4)-(4x^4-4)$
$=x^4(x^4-1)-4(x^4-1)=(x^4-1)(x^4-4)$
$=(x^2-1)(x^2+1)(x^2-2)(x^2+2)$
$=(x-1)(x+1)(x^2+1)(x-\sqrt{2})(x+\sqrt{2})(x^2+2)$
c.
$x^3+3x^2+3x-7=(x^3+3x^2+3x+1)-8$
$=(x+1)^3-2^3=(x+1-2)[(x+1)^2+2(x+1)+4]$
$=(x-1)(x^2+4x+7)$
a) \(3x^2+xy-4y^2=3x^2-3xy+4xy-4y^2\)
\(=3x(x-y)+4y(x-y)=(3x+4y)(x-y)\)
b)\(x^8-5x^4+4=x^8-x^4-4x^4+4\)
\(=x^2(x^4-1)-4(x^4-1)=(x^2-4)(x^4-1)\)
\(=(x-2)(x+2)(x^2-1)(x^2+1)=(x-2)(x+2)(x-1)(x+1)(x^2+1)\)
c)\(x^3+3x^2+3x-7=x^3+3x^2+3x+1-8\)
\(\left(x+1\right)^3-\sqrt{2}^3=\left(x+1-\sqrt[]{2}\right)\left(\left(x+1\right)^2+2\sqrt{2}x+2\right)\)
a: \(3x^2+xy-4y^2\)
\(=3x^2+4xy-3xy-4y^2\)
\(=x\left(3x+4y\right)-y\left(3x+4y\right)\)
\(=\left(3x+4y\right)\left(x-y\right)\)
b: \(x^8-5x^4+4\)
\(=x^8-x^4-4x^4+4\)
\(=x^4\left(x^4-1\right)-4\left(x^4-1\right)\)
\(=\left(x^4-4\right)\left(x^4-1\right)\)
\(=\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\left(x^2-2\right)\left(x^2+2\right)\)
phân tích các đa thức sau thành nhân tử:
a)3x-6x2y b)x3+22y+xy2-4x c) x2-6x+8
c: =(x-2)(x-4)
b: \(=x\left(x^2+2xy+y^2-4\right)\)
=x(x+y-2)(x+y+2)
Phân tích các đa thức sau thành nhân tử:
a) xy2-25x
b) x(x-y)2x-2y
c) x3-3x2-4x+12
a) \(xy^2-25x=x\left(y^2-25\right)=x\left(y-5\right)\left(y+5\right)\)
b) \(x\left(x-y\right)+2x-2y=x\left(x-y\right)+\left(2x-2y\right)=x\left(x-y\right)+2\left(x-y\right)=\left(x-y\right)\left(x+2\right)\)
c) \(x^3-3x^2-4x+12=\left(x^3-3x^2\right)-\left(4x-12\right)=x^2\left(x-3\right)-4\left(x-3\right)=\left(x-3\right)\left(x^2-4\right)=\left(x-2\right)\left(x-3\right)\left(x+2\right)\)
\(a,=x\left(y^2-25\right)=x\left(y-5\right)\left(y+5\right)\\ b,=x\left(x-y\right)+2\left(x-y\right)=\left(x+2\right)\left(x-y\right)\\ c,=x^2\left(x-3\right)-4\left(x-3\right)=\left(x-2\right)\left(x+2\right)\left(x-3\right)\)
\(x\left(y^2-25\right)=y\left(y-5\right)\left(y+5\right)\)
\(x\left(x-y\right)2\left(x-y\right)=\left(x-y\right)2x\)
\(x^2\left(x-3\right)-4\left(x-3\right)=\left(x^2-4\right)\left(x-3\right)=\left(x-2\right)\left(x+2\right)\left(x-3\right)\)
Phân tích các đa thức sau thành nhân tử:
a) x3+y3+x+y
b) x3−y3+x−y
c) (x−y)3+(x+y)3
d) x3−3x2y+3xy2−y3+y2−x2
`a, x^3 + y^3 + x + y`
`= (x+y)(x^2-xy+y^2)+x+y`
`= (x+y)(x^2-xy+y^2+1)`
`b, x^3 - y^3 + x -y`
`= (x-y)(x^2+xy+y^2)+x-y`
`= (x-y)(x^2+xy+y^2+1)`
`c, (x-y)^3 + (x+y)^3`
`= (x-y+x+y)(x^2-2xy+y^2 - x^2 + y^2 + x^2 + 2xy + y^2)`
`= (2x)(x^2 + 3y^2)`
`d, x^3 - 3x^2y + 3xy^2 - y^3 + y^2 - x^2`
`= (x-y)^3 + (y-x)(x+y)`
`=(x-y)(x^2+2xy+y^2-x-y)`
a: =(x+y)(x^2-xy+y^2)+(x+y)
=(x+y)(x^2-xy+y^2+1)
b: =(x-y)(x^2+xy+y^2)+(x-y)
=(x-y)(x^2+xy+y^2+1)
c: =x^3-3x^2y+3xy^2-y^3+x^3+3x^2y+3xy^2-y^3
=2x^3+6xy^2
d: =(x-y)^3+(y-x)(y+x)
=(x-y)[(x-y)^2-(x+y)]
Phân tích các đa thức sau thành nhân tử:
a) x3 – 2x2y + xy2
b) x2 + 12x + 20
c) (x2 + x + 1)(x2 + x + 4) + 2
a) \(=x\left(x^2-2xy+y^2\right)=x\left(x-y\right)^2\)
b) \(=\left(x^2+2x\right)+\left(10x+20\right)=x\left(x+2\right)+10\left(x+2\right)=\left(x+2\right)\left(x+10\right)\)
c) đặt \(x^2+x+1=t\)
\(\left(x^2+x+1\right)\left(x^2+x+4\right)+2=t\left(t+3\right)+2=t^2+3t+2=\left(t^2+t\right)+\left(2t+2\right)=t\left(t+1\right)+2\left(t+1\right)=\left(t+1\right)\left(t+2\right)=\left(x^2+x+2\right)\left(x^2+x+3\right)\)
Phân tích các đa thức sau thành nhân tử:
a) 3x4y - 12x2y3
b) x2 - y2 - 8y -16
c) x3 +3x2 + 4x +12
d) 3x2 - 6xy + 3y2 - 27
\(3x^4y-12x^2y^3=3x^2y\left(x^2-4y^2\right)=3x^2y\left(x-2y\right)\left(x+2y\right)\)
\(x^2-y^2-8y-16=x^2-\left(y^2+8y+16\right)=x^2-\left(y+4\right)^2=\left(x+y+4\right)\left(x-y-4\right)\)
\(x^3+3x^2+4x+12=x^2\left(x+3\right)+4\left(x+3\right)=\left(x^2+4\right)\left(x+3\right)\)
\(3x^2-6xy+3y^2-27=3\left[\left(x-y\right)^2-9\right]=3\left(x-y-3\right)\left(x-y+3\right)\)
Phân tích các đa thức sau thành nhân tử:
a) A= 4x3-8x2+4x
b) B= y2+x2-16-2xy
c) C= x3-8-3(2-x)
\(A=4x\left(x^2-2x+1\right)=4x\left(x-1\right)^2\\ B=\left(x-y\right)^2-16=\left(x-y-4\right)\left(x-y+4\right)\\ C=\left(x-2\right)\left(x^2+2x+4\right)+3\left(x-2\right)=\left(x-2\right)\left(x^2+2x+7\right)\)
a) \(A=4x\left(x^2-2x+1\right)=4x\left(x-1\right)^2\)
b) \(B=\left(x^2-2xy+y^2\right)-16=\left(x-y\right)^2-16=\left(x-y-4\right)\left(x-y+4\right)\)
c) \(C=\left(x-2\right)\left(x^2+2x+4\right)+3\left(x-2\right)=\left(x-2\right)\left(x^2+2x+7\right)\)
Phân tích các đa thức sau thành nhân tử:
a/ x2 – 3x – 4xy + 12y b/ x3 – 4x2 + 4x -1
c/ x – y – ax + ay d/ x2 – 4 + ( x + 2)2
e/x3 + x2y – x2z – xyz f/ x2 – y2 – 2x – 2y
a: \(=x\left(x-3\right)-4y\left(x-3\right)\)
=(x-3)(x-4y)
d: \(=\left(x-2\right)\left(x+2\right)+\left(x+2\right)^2\)
\(=\left(x+2\right)\left(x-2+x+2\right)\)
=2x(x+2)
\(a,=x\left(x-3\right)-4y\left(x-3\right)=\left(x-4y\right)\left(x-3\right)\\ b,=\left(x-1\right)\left(x^2+x+1\right)-4x\left(x-1\right)=\left(x-1\right)\left(x^2-3x+1\right)\\ c,=\left(x-y\right)\left(1-a\right)\\ d,=\left(x-2\right)\left(x-2+x+2\right)=2x\left(x-2\right)\\ e,=x^2\left(x+y\right)-xz\left(x+y\right)=x\left(x-z\right)\left(x+y\right)\\ f,=\left(x-y-2\right)\left(x+y\right)\)