Phân tích đa thức thành nhân tử:
a) x5 + x + 1
b) x4 + 2002x2 + 2001x + 2002
c) x11 + x7 + 1
Ai làm đúng + nhanh nhất mk tk
Phân tích đa thức thành nhân tử:
a) x4+4 b) x8+x7+1
c) x8+x4+1 d) x5+x+1
e) x2+2x2-24 f) a4+4b4
a: \(x^4+4=\left(x^2-2x+2\right)\left(x^2+2x+2\right)\)
b: \(x^8+x^7+1\)
\(=x^8+x^7+x^6-x^6-x^5-x^4+x^5+x^4+x^3-x^3-x^2-x+x^2+x+1\)
\(=\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)
c: \(x^8+x^4+1\)
\(=\left(x^8+2x^4+1\right)-x^4\)
\(=\left(x^4-x^2+1\right)\cdot\left(x^4+x^2+1\right)\)
\(=\left(x^4-x^2+1\right)\left(x^2+1-x\right)\left(x^2+1+x\right)\)
a)\(x^4+4\\ =\left(x^2\right)^2+4x^2+4-4x^2\\ =\left[\left(x^2\right)^2+4x^2+4\right]-\left(2x\right)^2\\ =\left(x^2+2\right)^2-\left(2x\right)^2\\ =\left(x^2+2+2x\right)\left(x^2+2-2x\right)\)
\(a)\; x^4+4 \\= x^4+4x^2+4-4x^2\\=(x^2+2)^2-4x^2\\=(x^2+2-2x)(x^2+2+2x)\)
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 các đa thức sau thành nhân tử:
a) A= \(x^3\)y - 12xy - x2y
b)B= 4x2 - 3y2 - 4xy - 2x + 3y
c)C= (x+1)(x+2)(x+3)(x+4) - 120
d)D= x5 - x4 + x2 - 1
a: \(A=x^3y-12xy-x^2y\)
\(=xy\cdot x^2-xy\cdot12-xy\cdot x\)
\(=xy\left(x^2-x-12\right)\)
\(=xy\left(x^2-4x+3x-12\right)\)
\(=xy\left[x\left(x-4\right)+3\left(x-4\right)\right]\)
\(=xy\left(x-4\right)\left(x+3\right)\)
c: \(C=\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-120\)
=(x+1)(x+4)(x+2)(x+3)-120
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-120\)
\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)+24-120\)
\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)-96\)
\(=\left(x^2+5x+16\right)\left(x^2+5x-6\right)\)
\(=\left(x^2+5x+16\right)\left(x+6\right)\left(x-1\right)\)
d: \(D=x^5-x^4+x^2-1\)
\(=\left(x^5-x^4\right)+\left(x^2-1\right)\)
\(=x^4\left(x-1\right)+\left(x-1\right)\left(x+1\right)\)
\(=\left(x-1\right)\left(x^4+x+1\right)\)
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: \(x^3y-2x^2y^2+5xy\)
\(=xy\left(x^2-2xy+5\right)\)
HELP ME!!!
Phân tích đa thức sau thành nhân tử bằng cách thêm bớt hạng tử, tách hạng tử
a, 6x2-11x
b, x7+x5+1
c, x8+x4+1
d, x3-5x+8-4
e, x5+x4+1
a. $6x^2-11x=x(6x-11)$
b. $x^7+x^5+1=(x^7-x)+(x^5-x^2)+x+x^2+1$
$=x(x^6-1)+x^2(x^3-1)+(x^2+x+1)$
$=x(x^3-1)(x^3+1)+x^2(x^3-1)+(x^2+x+1)$
$=(x^3-1)(x^4+x+x^2)+(x^2+x+1)$
$=(x-1)(x^2+x+1)(x^4+x^2+x)+(x^2+x+1)$
$=(x^2+x+1)[(x-1)(x^4+x^2+x)+1]$
$=(x^2+x+1)(x^5-x^4+x^3-x+1)$
c.
$x^8+x^4+1=(x^4)^2+2.x^4+1-x^4$
$=(x^4+1)^2-(x^2)^2$
$=(x^4+1-x^2)(x^4+1+x^2)$
$=(x^4+1-x^2)(x^4+2x^2+1-x^2)$
$=(x^4-x^2+1)[(x^2+1)^2-x^2]$
$=(x^4-x^2+1)(x^2+1-x)(x^2+1+x)$
d.
$x^3-5x+8-4=x^3-5x+4$
$=x^3-x^2+x^2-x-(4x-4)$
$=x^2(x-1)+x(x-1)-4(x-1)=(x-1)(x^2+x-4)$
e.
$x^5+x^4+1=(x^5-x^2)+(x^4-x)+x^2+x+1$
$=x^2(x^3-1)+x(x^3-1)+x^2+x+1$
$=(x^3-1)(x^2+x)+(x^2+x+1)$
$=(x-1)(x^2+x+1)(x^2+x)+(x^2+x+1)$
$=(x^2+x+1)[(x-1)(x^2+x)+1]$
$=(x^2+x+1)(x^3-x+1)$
Bài 1: Phân tích đa thức thành nhân tử:
a) x2y+xy+x+1
b) x2-(a+b)x+ab
c) ax2+ay-bx2-by
d) ax-2x-a2+2a
e) 2x2+4ax+x+2a
f) x3+ax2+x+a
g) x4+2x3-4x-4
a) x2y+xy+x+1= (x2y+xy)+(x+1)=xy(x+10+(x+1)=(x+1)(xy+1)
b) x2-(a+b)x+ab=x2-ax-bx+ab=(x2-ax)-(bx-ab)=x(x-a)-b(x-a)=(x-a)(x-b)
c) ax2+ay-bx2-by=(ax2+ay)-(bx2+by)=a(x2+y)-b(x2+y)=(a-b)(x2+y)
d) ax-2x-a2+2a=(ax-2x)-(a2-2a)=x(a-2)-a(a-2)=(a-2)(x-a)
e) 2x2+4ax+x+2a=(2x2+4ax)+(x+2a)=2x(x+2a)+(x+2a)=(x+2a)(2x+1)
f) x3+ax2+x+a=(x3+ax2)+(x+a)=x2(x+a)+(x+a)=(x2+1)(x+a)
g: Ta có: \(x^4+2x^3-4x-4\)
\(=\left(x^2-2\right)\left(x^2+2\right)-2x\left(x^2-2\right)\)
\(=\left(x^2-2\right)\cdot\left(x^2+2x+2\right)\)
Bài 1:phân tích đa thức thành nhân tử
a)x2-2x-4y2-4y e)x4+2x3+2x2+2x+1
b)x3+2x2+2x+1 f)x5+x4+x3+x2+x+1
c)x3-4x2+12x-27
d)a6-a4+2a3+2a2
Làm chi tiết giúp mình với ạ, cảm ơn
a) \(x^2-2x-4y^2-4y=\left(x^2-4y^2\right)-\left(2x+4y\right)=\left(x-2y\right)\left(x+2y\right)-2\left(x+2y\right)=\left(x+2y\right)\left(x-2y-2\right)\)
b) \(x^3+2x^2+2x+1=\left(x+1\right)\left(x^2-x+1\right)+2x\left(x+1\right)=\left(x+1\right)\left(x^2-x+1+2x\right)=\left(x+1\right)\left(x^2+x+1\right)\)
c) \(x^3-4x^2+12x-27=x^3-3x^2-x^2+3x+9x-27=x^2\left(x-3\right)-x\left(x-3\right)+9\left(x-3\right)=\left(x-3\right)\left(x^2-x+9\right)\)
d) \(a^6-a^4+2a^3+2a^2=a^2\left(a^4-a^2+2a+2\right)=a^2\left[a^2\left(a-1\right)\left(a+1\right)+2\left(a+1\right)\right]=a^2\left(a+1\right)\left(a^3-a^2+2\right)=a^2\left(a+1\right)\left[a^3+a^2-2a^2+2\right]=a^2\left(a+1\right)\left[a^2\left(a+1\right)-2\left(a-1\right)\left(a+1\right)\right]=a^2\left(a+1\right)^2\left(a^2-2a+2\right)\)
a) Ta có: \(x^2-2x-4y^2-4y\)
\(=\left(x^2-4y^2\right)-\left(2x+4y\right)\)
\(=\left(x-2y\right)\left(x+2y\right)-2\left(x+2y\right)\)
\(=\left(x+2y\right)\left(x-2y-2\right)\)
b) Ta có: \(x^3+2x^2+2x+1\)
\(=\left(x^3+1\right)+2x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-x+1\right)+2x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2+x+1\right)\)
d) Ta có: \(a^6-a^4+2a^3+2a^2\)
\(=a^2\left(a^4-a^2+2a+2\right)\)
\(=a^2\left[a^2\left(a^2-1\right)+\left(2a+2\right)\right]\)
\(=a^2\left[a^2\left(a-1\right)\left(a+1\right)+2\left(a+1\right)\right]\)
\(=a^2\cdot\left(a+1\right)\left(a^3-a+2\right)\)
c) Ta có: \(x^3-4x^2+12x-27\)
\(=\left(x^3-27\right)-\left(4x^2-12x\right)\)
\(=\left(x-3\right)\left(x^2+3x+9\right)-4x\left(x-3\right)\)
\(=\left(x-3\right)\left(x^2-x+9\right)\)
Phân tích đa thức thành nhân tử:
A= x4 + 4.
\(x^4+4\)
\(=x^4+4x^2+4-4x^2\)
\(=\left(x^2+2\right)^2-4x^2\)
\(=\left(x^2+2x+2\right)\left(x^2-2x+2\right)\)
\(A=x^4+4\)
\(=\) \(x^4+4+4x^2-4x^2\)
\(=\left(x^2+2\right)^2-4x^2\)
\(A=\) \(\left(x^2+2-2x\right)\left(x^2+2+2x\right)\)
phân tích đa thức thành nhân tử:a, x^2(x+1)-2x(x+1)+x+1b, 4x^2 - 8x +3
a) \(x^2 (x+1)-2x(x+1)+x+1 \\ =(x+1)(x^2-2x+1)\\=(x+1)(x-1)^2\)
b) \(4x^2 -8x+3 \\= (2x^2)-2.2x .2 + 2^2 -1 \\=(2x-2)^2-1^2\\=(2x-2+1)(2x-2-1)\\= (2x-1)(2x-3)\)