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\(x^3-4x^2-8x+8\)
\(\Leftrightarrow\left(x^3-4x^2\right)-\left(8x-8\right)\)
\(\Leftrightarrow x^2\left(x-4\right)-4\left(x-4\right)\)
\(\Leftrightarrow\left(x-4\right)\left(x^2-4\right)\)
1.\(x^2-2x-4y^2-4y=\left(x+2y\right)\left(x-2y\right)-2\left(x+2y\right)=\left(x+2y\right)\left(x-2y-2\right)\)
2.\(x^4+2x^3-4x-4=\left(x^2+2\right)\left(x^2-2\right)+2x\left(x^2-2\right)=\left(x^2-2\right)\left(x^2+2x-2\right)\)
3.\(3x^2-3y^2-2\left(x-y\right)^2=3\left(x-y\right)\left(x+y\right)-2\left(x-y\right)\left(x-y\right)=\left(x-y\right)\left(3x+3y-2x+2y\right)\)\(=\left(x-y\right)\left(x+5y\right)\)
4.\(x^3-4x^2-9x+36=x^2\left(x-4\right)-9\left(x-4\right)=\left(x-3\right)\left(x+3\right)\left(x-4\right)\)
5.\(\left(x-1\right)\left(2x+1\right)+3\left(x-1\right)\left(x+2\right)\left(2x+1\right)=\left(x-1\right)\left(2x+1\right)\left(1+3x+6\right)\)\(=\left(x-1\right)\left(2x+1\right)\left(3x+7\right)\)
6.\(\left(6x+3\right)-\left(2x-5\right)\left(2x+1\right)=3\left(2x+1\right)-\left(2x-5\right)\left(2x+1\right)\)\(=\left(2x+1\right)\left(3-2x-5\right)=\left(2x+1\right)\left(-2-2x\right)=-2\left(2x+1\right)\left(x+1\right)\)
7.\(\left(x-5\right)^2+\left(x+5\right)\left(x-5\right)+\left(x-5\right)\left(2x+1\right)=\left(x-5\right)\left(x-5+x+5+2x+1\right)\)\(=\left(x-5\right)\left(4x+1\right)\)
8.\(\left(3x-2\right)\left(4x-3\right)+\left(3x-2\right)\left(x-1\right)-2\left(3x-2\right)\left(x+1\right)\)\(=\left(3x-2\right)\left(4x-3+x-1-2x-2\right)=\left(3x-2\right)\left(3x-6\right)=3\left(3x-2\right)\left(x-2\right)\)
+,
= (x-y)^2 - z.(x-y) = (x-y).(x-y-z)
+,
=(x-y).(x+y)-(x-y) = (x-y).(x+y-1)
+,
=x^3.(x-1)-(x^2-1) = x^3.(x-1).(x-1).(x+1) = (x-1).(x^3-x-1)
+,
=a.(x^2-y^2)-7.(x+y) = a.(x+y).(x-y)-7.(x+y) = (ax+ay-7).(x+y)
\(x^2-2xy+y^2-xz+yz\)
\(=\left(x-y\right)^2-\left(xz-yz\right)\)
\(=\left(x-y\right)\left(x-y\right)-z\left(x-y\right)\)
\(=\left(x-y\right)\left(x-y-z\right)\)
\(x^2-y^2-x+y\)
\(=\left(x-y\right)\left(x+y\right)-\left(x-y\right)\)
\(=\left(x-y\right)\left(x+y-1\right)\)
\(x^4-x^3-x^2+1\)
\(=x^3\left(x-1\right)-\left(x^2-1\right)\)
\(=x^3\left(x-1\right)-\left(x-1\right)\left(x+1\right)\)
\(=\left(x-1\right)\left(x^3-x-1\right)\)
\(ax^2-ay^2-7x-7y\)
\(=a\left(x^2-y^2\right)-\left(7x+7y\right)\)
\(=a\left(x-y\right)\left(x+y\right)-7\left(x-y\right)\)
\(=\left(x-y\right)\left[a\left(x+y\right)-7\right]\)
x4+x=x(x3+1)=x(x+1)(x2-x+1)
x4+64=x4+16x2+64-16x2=(x2+8)2-(4x)2=(x2+8+4x)(x2+8-4x)
4x4+81=4x4+36x2+81-36x2=(2x2+9)2-(6x)2=(2x2+9+6x)(2x2+9-6x)
64x4+y4=64x4+16(xy)2+y4-16(xy)2=(8x2+y2)-(4xy)2=(8x2+y2-4xy)(8x2+y2=4xy)
x4+4y4=x4+4(xy)2+4y4-4(xy)2=(x2+2y2-2xy)(x2+2y2+2xy)
x4+x2+1=(x4+2x2+1)-x2=(x2+1-x)(x2+1+x)
Mình làm có vài đoạn hơi tắt nha.
\(a,3x^2+2x-1\)
\(\Leftrightarrow3x^2+3x-x-1\)
\(\Leftrightarrow3x\left(x+1\right)-\left(x+1\right)\)
\(\Leftrightarrow\left(x+1\right)\left(3x-1\right)\)
\(b,x^3+6x^2+11x+6\)
\(\Leftrightarrow x^3+3x^2+3x^2+9x+2x+6\)
\(\Leftrightarrow x^2\left(x+3\right)+3x\left(x+3\right)+2\left(x+6\right)\)
\(\Leftrightarrow\left(x+3\right)\left(x^2+3x+2\right)\)
\(\Leftrightarrow\left(x+3\right)\left(x+1\right)\left(x+2\right)\)
\(c,x^4+2x^2-3\)
\(\Leftrightarrow x^4-x^3+x^3-x^2+3x^2-3\)
\(\Leftrightarrow x^3\left(x-1\right)+x^2\left(x-1\right)+3\left(x-1\right)\left(x+1\right)\)
\(\Leftrightarrow\left(x-1\right)\left(x^3+x^2+3x+3\right)\)
\(\Leftrightarrow\left(x-1\right)\left(x+1\right)\left(x+3\right)\)
\(d,ab+ac+b^2+2bc+c^2\)
\(\Leftrightarrow a\left(b+c\right)+\left(b+c\right)^2\)
\(\Leftrightarrow\left(b+c\right)\left(a+b+c\right)\)
3x^2+2x-1=3x^2+3x-x-1=3x(x+1)-(x+1)=(x+1)(3x-1)
x^4+2x^2-3=x^4+3x^2-x^2 -3=x^2(x^2+3)-(x^2+3)=(x^2+3)(x^2-1)
\(1,\)\(3x-3y-x^2+2xy-y^2\)
\(=-\left(x^2-2xy+y^2\right)+\left(3x-3y\right)\)
\(=-\left(x-y\right)^2+3\left(x-y\right)\)
\(=\left(x-y\right)\left[-\left(x-y\right)+3\right]\)
\(=\left(x-y\right)\left(3-x+y\right)\)
\(2,\)\(49\left(x-4\right)^2-9\left(x+2\right)^2\)
\(=\left[7\left(x-4\right)\right]^2-\left[3\left(x-2\right)\right]^2\)
\(=\left(7x-28-3x+6\right)\left(7x-28+3x-6\right)\)
\(=\left(4x-22\right)\left(10x+34\right)\)
\(3,\)\(x^4+4x^2-5\)
\(=x^4-x^2+5x^2-5\)
\(=x^2\left(x^2-1\right)+5\left(x^2-1\right)\)
\(=\left(x^2+5\right)\left(x-1\right)\left(x+1\right)\)
x2-7x+12
=x2-3x-4x+12
=x(x-3)-4(x-3)
=(x-3)(x-4)
x4-4x2+4x-1
=x4-1-4x2+4x
=(x2-1)(x2+1)-4x(x-1)
=(x-1)(x+1)(x2+1)-4x(x-1)
=(x-1)[(x+1)(x2+1)-4x]
=(x-1)(x3+x2+x+1-4x)
=(x-1)(x3+x2-3x+1)
6x4-11x2+3
=6x4-2x2-9x2+3
=2x2(3x2-1)-3(3x2-1)
=(3x2-1)(2x2-3)