Ta có:

A = (x – y).(x2 + xy + y2)

   = x.(x2 + xy + y2) + (–y).(x2 + xy + y2)

   = x.x2 + x.xy + x.y2 + (–y).x2 + (–y).xy + (–y).y2

   = x3 + x2y + xy2 – x2y – xy2 – y3

   = x3 – y3 + (x2y – x2y) + (xy2 – xy2)

   = x3 – y3.

Tại x = –10, y = 2 thì A = (–10)3 – 23 = –1000 – 8 = –1008

Tại x = –1 ; y = 0 thì A = (–1)3 – 03 = –1 – 0 = –1

Tại x = 2 ; y = –1 thì A = 23 – (–1)3 = 8 – (–1) = 9

Tại x = –0,5 ; y = 1,25 thì A = (–0,5)3 – 1,253 = –0,125 – 1,953125 = –2,078125

Vậy ta có bảng sau :

\(P=\left(x+y\right)\left\{\left[\left(x+y\right)^2-2xy\right]\left[\left(x+y\right)^3-3xy\left(x+y\right)\right]\right\}\\ \)

Thây số vào

VÌ \(x+y=7;xy=10\)

\(\Rightarrow x,y=5\)và \(2\)

\(\Rightarrow P=\left(5+2\right)\left(5^2+2^2\right)\left(5^3+2^3\right)\)

\(\Rightarrow P=7.29.133\)

    \(P=26999\)

26999

Lời giải:

$P=(xy+yz+xz)^2+(x^2-yz)^2+(y^2-zx)^2+(z^2-xy)^2$
$=x^2y^2+y^2z^2+z^2x^2+2x^2yz+2xy^2z+2xyz^2+x^4+y^2z^2-2x^2yz+y^4+z^2x^2-2xzy^2+z^4+x^2y^2-2xyz^2$

$=x^4+y^4+z^4+2x^2y^2+2y^2z^2+2z^2x^2$

$=(x^2+y^2+z^2)^2=10^2=100$

ms lm xong luon này

\(xy+\sqrt{\left(1+x^2\right)\left(1+y^2\right)}=a\)

\(\Rightarrow x^2y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}+\left(1+x^2\right)\left(1+y^2\right)=a^2\)

\(\Rightarrow x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+2.x\sqrt{1+y^2}.y\sqrt{1+x^2}+1=a^2\)

\(\Rightarrow\left(x\sqrt{1+y^2}+y\sqrt{1+x^2}\right)^2+1=a^2\)

\(\Rightarrow E^2+1=a^2\)

\(\Rightarrow E=\pm\sqrt{a^2-1}\)

\(a^2=x^2y^2+(1+x^2)(1+y^2)+2xy\sqrt{(1+x^2)(1+y^2)} \\->2xy\sqrt{(1+x^2)(1+y^2)}=a^2-2x^2y^2-1-x^2-y^2 \\E^2=x^2(1+y^2)+y^2(1+x^2)+2xy\sqrt{(1+x^2)(1+y^2)} \\=x^2+y^2+2x^2y^2+a^2-2x^2y^2-1-x^2-y^2 \\=a^2-1\)

\(E^2=x^2\left(y^2+1\right)+y^2\left(x^2+1\right)+2xy\sqrt{\left(y^2+1\right)\left(x^2+1\right)}\)

\(=2\left(xy\right)^2+x^2+y^2+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}\)

\(a^2=\left(xy\right)^2+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}+\left(x^2+1\right)\left(y^2+1\right)\)

\(=2\left(xy\right)^2+2xy\sqrt{\left(x^2+1\right)\left(y^2+1\right)}+x^2+y^2+1\)

\(\Rightarrow E^2=a^2-1\Rightarrow E=\sqrt{a^2-1}\)

\(E=x\sqrt{1+y^2}+y\sqrt{1+x^2}\)

\(\Leftrightarrow E^2=x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\)

\(=2x^2y^2+x^2+y^2+2xy\left(a-xy\right)\)

\(=2x^2y^2+x^2+y^2+2xya-2x^2y^2\)

\(=x^2+y^2+2xya\)

\(=\left(2xy\right)2+a=a^2+a=E^2\)

\(E=\sqrt{a^2+a}\)

\(\rightarrow E^2=x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+\\ 2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\\ =2xy^2+x^2+y^2+2xy\left(a-xy\right)\\ =2x^2y^2+x^2+y^2+2xya-2x^2y^2\\ =x^2+y^2+2xya\\ =\left(x+y\right)^2+a=a^2+a\\ =E^2\\ Vậy.E=\sqrt{a^2+a}\)