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`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)]
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\(=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left(x^2+2xy+y^2-1\right)\)
\(x^3-x+3x^2+3xy^2+y^3-y\)
\(=\left(x+y\right)^3-\left(x+y\right)\)
\(=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
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x 3 - x + 3 x 2 y + 3 x y 2 + y 3 - y = x 3 + 3 x 2 y + 3 x y 2 + y 3 - x - y = x + y 3 - x - y = x + y x + y 2 - 1 = x + y x + y + 1 x + y - 1
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Ta có
N = x 3 + 3 x 2 y + 3 x y 2 + y 3 + x 2 + 2 x y + y 2 = ( x 3 + 3 x 2 y + 3 x y 2 + y 3 ) + ( x 2 + 2 x y + y 2 ) = ( x + y ) 3 + ( x + y ) 2 = ( x + y ) 2 ( x + y + 1 )
Từ đề bài x = 10 – y ó x + y = 10. Thay x + y = 10 vào N = ( x + y ) 2 (x + y + 1) ta được
N = 10 2 (10 + 1) = 1100
Suy ra N > 1000 khi x = 10 – y
Đáp án cần chọn là: D
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c) \(3x+3y-x^2-2xy-y^2=3\left(x+y\right)-\left(x+y\right)^2=\left(x+y\right)\left(3-x-y\right)\)d) \(=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left[\left(x+y\right)^2-1\right]\)
\(=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
\(c,=3\left(x+y\right)-\left(x+y\right)^2=\left(3-x-y\right)\left(x+y\right)\\ d,=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left[\left(x+y\right)^2-1\right]\\ =\left(x+y\right)\left(x+y-1\right)\left(x+y+1\right)\)