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a: \(x^2-y^2-x-y\)
\(=\left(x-y\right)\left(x+y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-1\right)\)
f: \(x^3-5x^2-5x+1\)
\(=\left(x+1\right)\left(x^2-x+1\right)-5x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-6x+1\right)\)
a: Ta có: \(\left(x+y\right)^2\)
\(=x^2+2xy+y^2\)
\(\Leftrightarrow x^2+y^2=\dfrac{\left(x+y\right)^2}{2xy}\ge\dfrac{\left(x+y\right)^2}{2}\forall x,y>0\)
\(\left(x+y-z\right)^3-x^3-y^3+z^3\)
\(=\left[\left(x+y\right)-z\right]^3-x^3-y^3+z^3\)
\(=\left(x+y\right)^3-z^3-3\left(x+y\right)z\left(x+y-z\right)-x^3-y^3+z^3\)
\(=x^3+y^3-z^3+3xy\left(x+y\right)-3\left(x+y\right)z\left(x+y-z\right)-x^3-y^3+z^3\)
\(=3xy\left(x+y\right)-3z\left(x+y\right)\left(x+y-z\right)\)
\(=3\left(x+y\right)\left[xy-z\left(x+y-z\right)\right]\)
\(=3\left(x+y\right)\left(xy-zx-yz+z^2\right)\)
\(=3\left(x+y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)
\(=3\left(x+y\right)\left(y-z\right)\left(x-z\right)\)
#\(Urushi\text{☕}\)
Áp dụng (a+b)3 = a3+b3+3ab(a+b), ta có:
(x+y+z)3-x3-y3-z3
=[(x+y)+z]3-x3-y3-z3
=(x+y)3+z3+3z(x+y)(x+y+z)-x3-y3-z3
=x3+y3+3xy(x+y)+z3+3z(x+y)(x+y+z)-x3-y3-z3
=3(x+y)(xy+xz+yz+z2)
=3(x+y)[x(y+z)+z(y+z)]
=3(x+y)(y+z)(x+z)
Ta có: ( x - y) z3 + ( y - z ) x3 + ( z - x ) y3
= ( x - y ) z3 + ( y - z )x3 + ( z - y)y3 + ( y - x ) y3
= ( x - y ) ( z3 - y3 ) + ( y - z ) ( x3 - y3)
= ( x - y ) ( z - y ) ( z2 + zy + y2 ) + ( y - z ) ( x - y) ( x2 + xy + y2 )
= ( x - y ) ( y - z ) ( x2 + xy + y2 - z2 - zy - y2)
= ( x - y ) ( y - z ) [ ( x2 - z2) + ( xy - zy) ]
= ( x - y ) ( y - z ) [ ( x - z ) ( x + z ) + y ( x - z ) ]
= ( x - y ) ( y - z ) ( x - z ) ( x + y + z )
(x - y).z3 + (y - z).x3 + (z - x).y3
= z3(x - y) + x3y - x3z + y3z - xy3
= z3(x - y) + xy(x2 - y2) - z(x3 - y3)
= z3(x - y) + xy(x - y)(x + y) - z(x - y)(x2 + xy + y2)
= (x - y)(z3 + x2y + xy2 - x2z - xyz - y2z)
= (x - y)[z(z2 - x2) + xy(x - z) + y2(x - z)]
= (x - y)[z(z - x)(z + x) - xy(z- x) - y2(z - x)]
= (x - y)(z - x)(z2 + xz - xy - y2)
= (x - y)(z - x)[(y - z)(y + z) - x(y - z)]
= (x - y)(z - x)(y - z)(y + z - x)
a) \(\left(x+2y\right)^2-\left(x-y\right)^2=\left(x+2y+x-y\right)\left(x+2y-x+y\right)\)
\(=\left(2x+y\right).3y\)
b) \(\left(x+1\right)^3+\left(x-1\right)^3\)
\(=\left(x+1+x-1\right)\left[\left(x+1\right)^2-\left(x+1\right)\left(x-1\right)+\left(x-1\right)^2\right]\)
\(=2x\left[\left(x+1\right)^2-\left(x^2-1\right)+\left(x-1\right)^2\right]\)
c) \(9x^2-3x+2y-4y^2\)
\(=9x^2-4y^2-3x+2y\)
\(=\left(3x-2y\right)\left(3x+2y\right)-\left(3x-2y\right)\)
\(=\left(3x-2y\right)\left[3x+2y-1\right]\)
d) \(4x^2-4xy+2x-y+y^2\)
\(=4x^2-4xy+y^2+2x-y\)
\(=\left(2x-y\right)^2+2x-y\)
\(=\left(2x-y\right)\left(2x-y+1\right)\)
e) \(x^3+3x^2+3x+1-y^3\)
\(=\left(x+1\right)^3-y^3\)
\(=\left(x+1-y\right)\left[\left(x+1\right)^2+y\left(x+1\right)+y^2\right]\)
g) \(x^3-2x^2y+xy^2-4x\)
\(=x\left(x^2-2xy+y^2\right)-4x\)
\(=x\left(x-y\right)^2-4x\)
\(=x\left[\left(x-y\right)^2-4\right]\)
\(=x\left(x-y+2\right)\left(x-y-2\right)\)
a) (x + 2y)² - (x - y)²
= (x + 2y - x + y)(x + 2y + x - y)
= 3y(2x + y)
b) (x + 1)³ + (x - 1)³
= (x + 1 + x - 1)[(x + 1)² - (x + 1)(x - 1) + (x - 1)²]
= 2x(x² + 2x + 1 - x² + 1 + x² - 2x + 1)
= 2x(x² + 3)
c) 9x² - 3x + 2y - 4y²
= (9x² - 4y²) - (3x - 2y)
= (3x - 2y)(3x + 2y) - (3x - 2y)
= (3x - 2y)(3x + 2y - 1)
d) 4x² - 4xy + 2x - y + y²
= (4x² - 4xy + y²) + (2x - y)
= (2x - y)² + (2x - y)
= (2x - y)(2x - y + 1)
e) x³ + 3x² + 3x + 1 - y³
= (x³ + 3x² + 3x + 1) - y³
= (x + 1)³ - y³
= (x + 1 - y)[(x + 1)² + (x + 1)y + y²]
= (x - y + 1)(x² + 2x + 1 + xy + y + y²)
g) x³ - 2x²y + xy² - 4x
= x(x² - 2xy + y² - 4)
= x[(x² - 2xy + y²) - 4]
= x[(x - y)² - 2²]
= x(x - y - 2)(x - y + 2)
a) x⁴ + 2x² + 1
= (x²)² + 2.x².1 + 1²
= (x² + 1)²
b) 4x² - 12xy + 9y²
= (2x)² - 2.2x.3y + (3y)²
= (2x - 3y)²
c) -x² - 2xy - y²
= -(x² + 2xy + y²)
= -(x + y)²
d) (x + y)² - 2(x + y) + 1
= (x + y)² - 2.(x + y).1 + 1²
= (x - y + 1)²
e) x³ - 3x² + 3x - 1
= x³ - 3.x².1 + 3.x.1² - 1³
= (x - 1)³
g) x³ + 6x² + 12x + 8
= x³ + 3.x².2 + 3.x.2² + 2³
= (x + 2)³
h) x³ + 1 - x² - x
= (x³ + 1) - (x² + x)
= (x + 1)(x² - x + 1) - x(x + 1)
= (x + 1)(x² - x + 1 - x)
= (x + 1)(x² - 2x + 1)
= (x + 1)(x - 1)²
k) (x + y)³ - x³ - y³
= (x + y)³ - (x³ + y³)
= (x + y)³ - (x + y)(x² - xy + y²)
= (x + y)[(x + y)² - x² + xy - y²]
= (x + y)(x² + 2xy + y² - x² + xy - y²)
= (x + y).3xy
= 3xy(x + y)
a: Ta có: \(x^2-4y^2-2x-4y\)
\(=\left(x-2y\right)\left(x+2y\right)-2\left(x+2y\right)\)
\(=\left(x+2y\right)\left(x-2y-2\right)\)
c: Ta có: \(x^3+2x^2y-x-2y\)
\(=x^2\left(x+2y\right)-\left(x+2y\right)\)
\(=\left(x+2y\right)\left(x-1\right)\left(x+1\right)\)
d: Ta có: \(3x^2-3y^2-2\cdot\left(x-y\right)^2\)
\(=3\left(x-y\right)\left(x+y\right)-2\cdot\left(x-y\right)^2\)
\(=\left(x-y\right)\left(3x+3y-2x+2y\right)\)
\(=\left(x-y\right)\left(x+5y\right)\)
e: Ta có: \(x^3-4x^2-9x+36\)
\(=x^2\left(x-4\right)-9\left(x-4\right)\)
\(=\left(x-4\right)\left(x-3\right)\left(x+3\right)\)
f: Ta có: \(x^2-y^2-2x-2y\)
\(=\left(x-y\right)\left(x+y\right)-2\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-2\right)\)
x4 - 5x2 +4
=x4 -4x -x +4
=(x4 - 4x) - (x-4)
=x3(x-4)-(x -4)
=(x3 -1).(x-4)
3, x3 - x + y3 - y
=( x3 + y3) - (x + y)
=(x +y)(x2 - xy +y2) -(x +y)
= x2 - xy + y2
4, x3 - 3x2 -4x +12
= (x3 - 3x2)-(4x -12)
=x2(x - 3) - 4(x - 3)
= (x2 - 4)(x -3)
=(x-2)(x+2)(x-3)
7, 45 +x3 - 5x2 - 9x
=( x3 - 5x2 )-( 9x - 45)
= x2(x-5)-9(x-5)
=( x2- 9)(x-5)
= (x-3)(x+3)(x-5)