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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
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) \(=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left[\left(x+y\right)^2-1\right]\)
\(=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
\(c,=3\left(x+y\right)-\left(x+y\right)^2=\left(3-x-y\right)\left(x+y\right)\\ d,=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left[\left(x+y\right)^2-1\right]\\ =\left(x+y\right)\left(x+y-1\right)\left(x+y+1\right)\)
a) Ta có: \(M=x^2-2xy+y^2-10x+10y\)
\(=\left(x-y\right)^2-10\left(x-y\right)\)
\(=9^2-10\cdot9=-9\)
\(=\left(x+y\right)^3-\left(x+y\right)=\left(x+y\right)\left(x^2+2xy+y^2-1\right)\)
\(x^3-x+3x^2+3xy^2+y^3-y\)
\(=\left(x+y\right)^3-\left(x+y\right)\)
\(=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
\(P=x^3-3x^2+3x^2y+3xy^2+y^3-3y^2-6xy+3x+3y+2015\)
\(\Leftrightarrow P=\left(x^3+3x^2y+3xy^2+y^3\right)-\left(3x^2+6xy+3y^2\right)+\left(3x+3y\right)+2015\)
\(\Leftrightarrow P=\left(x+y\right)^3-3\left(x^2+2xy+y^2\right)+3\left(x+y\right)+2015\)
\(\Leftrightarrow P=\left(x+y\right)^3-3\left(x+y\right)^2+3\left(x+y\right)+2015\)
\(\Leftrightarrow P=101^3-3.101^2+3.101+2015\)
\(P=x^3-3x^2+3x^2y+3xy^2+y^3-3y^2-6xy+3x+3y+2015\)
\(\Leftrightarrow P=x^3+3x^2y+3xy^2+y^3-3x^2-6xy-3y^2+3x+3y+2015\)
\(\Leftrightarrow P=\left(x^3+3x^2y+3xy^2+y^3\right)-\left(3x^2+6xy+3y^2\right)+\left(3x+3y\right)+2015\)
\(\Leftrightarrow P=\left(x^3+3x^2y+3xy^2+y^3\right)-3\left(x^2+2xy+y^2\right)+3\left(x+y\right)+2015\)
\(\Leftrightarrow P=\left(x+y\right)^3-3\left(x+y\right)^2+3\left(x+y\right)+2015\)
\(\Leftrightarrow P=101^3-3.101^2+3.101+2015\)
\(\Leftrightarrow P=1030301-30603+303+2015\)
\(\Leftrightarrow P=999698+303+2015\)
\(\Leftrightarrow P=1000001+2015\)
\(\Leftrightarrow P=1002016\)