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x^4-5x^2+4=x^4-x^2-(4x^2-4) = x^2(x^2-1)-4(x^2-1)
=(x^2-4)(x^2-1)
=(x-2)(x+2)(x-1)(x+1)
4. 4x2 + 4x + 1 = ( 2x + 1)2
5. \(\dfrac{1}{4}x-\dfrac{2}{3}xy+\dfrac{4}{9}y^2\) \(=\left(\dfrac{1}{2}x\right)^2-2.\dfrac{1}{2}x.\dfrac{2}{3}+\left(\dfrac{2}{3}y\right)^2\)
\(=\left(\dfrac{1}{2}x-\dfrac{2}{3}y\right)^2\)
6. \(4a^2-\dfrac{4}{3}ab+\dfrac{1}{9}b^2=\left(2a\right)^2-2.2a.\dfrac{1}{3}+\left(\dfrac{1}{3}b\right)^2=\left(2a-\dfrac{1}{3}b\right)^2\)
7.
\(9x^2+4xy+\dfrac{4}{9}y^2-25z^2=\left(3x+\dfrac{2}{3}y\right)^2-\left(5z\right)^2=\left(3x+\dfrac{2}{3}y-5z\right)\left(3x+\dfrac{2}{3}y+5z\right)\)
1. \(x^4-2x^2+1=\left(x^2-1\right)^2\)
2. \(x^2+5x+\dfrac{25}{4}=x^2+2.x.\dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2=\left(x+\dfrac{5}{2}\right)^2\)
3. \(16x^2-8x+1=\left(4x-1\right)^2\)
4. \(x^2+x-y^2+y=\left(x-y\right)\left(x+y\right)+\left(x+y\right)=\left(x-y+1\right)\left(x+y\right)\)
5. \(\dfrac{1}{4}x^2-\dfrac{4}{9}y^2=\left(\dfrac{1}{2}x-\dfrac{2}{3}y\right)\left(\dfrac{1}{2}x+\dfrac{2}{3}y\right)\)
6. \(a^2-2ab+b^2-x^2=\left(a-b\right)^2-x^2=\left(a-b-x\right)\left(a-b+x\right)\)
7. \(4x^2-20x+25-y^2=\left(2x-5\right)^2-y^2=\left(2x-5-y\right)\left(2x-5+y\right)\)
h, \(27x^3-8=\left(3x-2\right)\left(9x^2+6x+4\right)\)
\(\Rightarrow\left(27x^3-8\right):\left(3x-2\right)\\ =\left(3x-2\right)\left(9x^2+6x+4\right):\left(3x-2\right)\\ =9x^2+6x+4\)
g, \(x^4-2x^2+1=\left(x^2-1\right)^2\)
\(\Rightarrow\left(x^4-2x^2+1\right):\left(1-x^2\right)\\ =\left(x^2-1\right)^2:\left(1-x^2\right)\\ =x^2-1\)
1 + 2xy - x2 - y2
= 1 - ( x2 - 2xy + y2 )
= 12 - ( x - y )2
= [ 1 - ( x - y ) ][ 1 + ( x - y ) ]
= ( y - x + 1 )( x - y + 1 )
a2 + b2 - c2 - d2 - 2ab + 2cd
= ( a2 - 2ab + b2 ) - ( c2 - 2cd + d2 )
= ( a - b )2 - ( c - d )2
= [ ( a - b ) - ( c - d ) ][ ( a - b ) + ( c - d ) ]
= ( a - b - c + d )( a - b + c - d )
a3b3 - 1
= ( ab )3 - 13
= ( ab - 1 )[ ( ab )2 + ab.1 + 12 ]
= ( ab - 1 )( a2b2 + ab + 1 )
x2( y - z ) + y2( z - x ) + z2( x - y )
= z2( x - y ) + x2y - x2z + y2z + y2x
= z2( x - y ) + ( x2y - y2x ) - ( x2z - y2z )
= z2( x - y ) + xy( x - y ) - z( x2 - y2 )
= z2( x - y ) + xy( x - y ) - z( x + y )( x - y )
= ( x - y )[ z2 + xy - z( x + y ) ]
= ( x - y )( z2 + xy - zx - zy )
= ( x - y )[ ( z2 - zx ) - ( zy - xy ) ]
= ( x - y )[ z( z - x ) - y( z - x ) ]
= ( x - y )( z - x )( z - y )
\(a,x^2+4x-y^2+4\)
\(=\left(x^2+4x+4\right)-y^2\)
\(=\left(x+2\right)^2-y^2\)
\(=\left(x+2-y\right)\left(x+2+y\right)\)
\(b,25-4x^2-4xy-y^2\)
\(=25-\left(4x^2+4xy+y^2\right)\)
\(=5^2-\left(2x+y\right)^2\)
\(=\left(5-2x+y\right)\left(5+2x+y\right)\)
\(c,x^3-x+y^3-y\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2-xy+y^2+1\right)\)
\(1,\)
\(\left(x^2-9y^2\right)\left(4x+12y\right)\)
\(=\left(x-3y\right)\left(x+3y\right)-4\left(x+3y\right)\)
\(=\left(x+3y\right)\left(x-3y-4\right)\)
\(3,\)
\(-x^2+2xy-y^2+25\)
\(=-\left(x^2-2xy+y^2\right)+25\)
\(=25-\left(x-y\right)^2\)
\(=5^2-\left(x-y\right)^2\)
\(=\left(5-x+y\right)\left(5+x-y\right)\)
Giải:
1) \(\left(x^2-y\right)^3\)
\(=x^6-3x^4y+4x^2y^2-y^3\)
Vậy ...
2) \(\left(x-2+y\right)^3\)
\(=\left(x-2\right)^3+3\left(x-2\right)^2y+3\left(x-2\right)y^2+y^3\)
\(=x^3-3x^2+16x-2^3+3\left(x^2-4x-4\right)y+3\left(x-2\right)y^2+y^3\)
\(=x^3-3x^2+16x-2^3+3x^2-12x-12y+3\left(xy^2-2y^2\right)+y^3\)
\(=x^3-3x^2+16x-2^3+3x^2-12x-12y+3xy^2-6y^2+y^3\)
\(=x^3+4x-8-12y+3xy^2-6y^2+y^3\)
Vậy ...
3) \(\left(z+y^2\right)^3\)
\(=z^3+3z^2y^2+3zy^4+y^6\)
Vậy ...
4) \(\left(x-y+z\right)^3\)
\(=\left(x-y\right)^3+3\left(x-y\right)^2z+3\left(x-y\right)z^2+z^3\)
\(=x^3-3x^2y+3xy^2-y^3+3\left(x^2-2xy+y^2\right)z+3\left(xz^2-yz^2\right)+z^3\)
\(=x^3-3x^2y+3xy^2-y^3+3x^2-6xy+3y^2z+3xz^2-3yz^2+z^3\)
\(=-3x^2y+3xy^2-y^3+4x^2-6xy+3y^2z+3xz^2-3yz^2+z^3\)
Vậy ...