Câu hỏi

C h o x , y ∈ Z c h ứ n g minh r a ˘ ˋ n g : N = ( x – y ) ( x –2 y ) ( x –3 y ) ( x –4 y ) + y 4 l a ˋ s o ^ ˊ c h ı ˊ nh p h ươ n g .

$C h o x, y \in Z c h ứ n g minh r \overset{ˋ}{\overset{a}{˘}} n g : N = (x - y) (x -2 y) (x -3 y) (x -4 y) + y^{4} l \overset{a}{ˋ} s \overset{ˊ}{\overset{o}{^}} c h \overset{ı}{ˊ} nh p h ươ n g .$

R. Roboteacher12

Giáo viên

University of Pedagogy

Xác nhận câu trả lời

Giải thích

N = ( x − y ) ( x − 4 y ) ( x − 2 y ) ( x − 3 y ) + y 4 = ( x 2 − 4 x y − x y + 4 y 2 ) ( x 2 − 3 x y − 2 x y + 6 y 2 ) + y 4 = ( x 2 − 5 x y + 4 y 2 ) ( x 2 − 5 x y + 6 y 2 ) + y 4 Đ ạ t t = x 2 − 5 x y + 4 y 2 , t a đư ợ c N = t ( t + 2 y 2 ) + y 4 = t 2 + 2 t y 2 + y 4 = ( t + y 2 ) 2 = ( x 2 − 5 x y + 4 y 2 + y 2 ) 2 = ( x 2 − 5 x y + 5 y 2 ) 2 m a ˋ x , y ∈ Z n e ^ n N l u o ^ n l a ˋ s o ^ ˊ c h ı ˊ nh p h ươ n g .

$N = (x - y) (x - 4 y) (x - 2 y) (x - 3 y) + y^{4} = (x^{2} - 4 x y - x y + 4 y^{2}) (x^{2} - 3 x y - 2 x y + 6 y^{2}) + y^{4} = (x^{2} - 5 x y + 4 y^{2}) (x^{2} - 5 x y + 6 y^{2}) + y^{4} Đ ạ t t = x^{2} - 5 x y + 4 y^{2}, t a đư ợ c N = t (t + 2 y^{2}) + y^{4} = t^{2} + 2 t y^{2} + y^{4} = (t + y^{2})^{2} = (x^{2} - 5 x y + 4 y^{2} + y^{2})^{2} = (x^{2} - 5 x y + 5 y^{2})^{2} m \overset{a}{ˋ} x, y \in Z n \overset{e}{^} n N l u \overset{o}{^} n l \overset{a}{ˋ} s \overset{ˊ}{\overset{o}{^}} c h \overset{ı}{ˊ} nh p h ươ n g .$