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Ta có \(x^3+y^3+z^3=3xyz\)
\(\Leftrightarrow\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz=0\)
\(\Leftrightarrow\left(x+y+z\right)\left(x^2+y^2+z^2+2xy-xz-yz\right)-3xy\left(x+y+z\right)=0\)
\(\Leftrightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)
\(\Leftrightarrow\left(x+y+z\right)\left[\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(z^2-2zx+x^2\right)\right]=0\)(Nhân hai vế với 2)
\(\Leftrightarrow\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]=0\)
Tới đây bạn xét hai trường hợp nhé :)
(x+y+z)((X+Y)^2-Z(X+Y))-3XY(X+Y+Z)
=(X+Y+Z)(X^2+2XY+Y^2-XZ-YZ-3XY)
=(X+Y+Z)(X^2+Y^2+Z^2-XZ-YZ-XY)
a) Ta có:
x³ + y³ + z³ - 3xyz = (x+y)³ - 3xy(x-y) + z³ - 3xyz
= [(x+y)³ + z³] - 3xy(x+y+z)
= (x+y+z)³ - 3z(x+y)(x+y+z) - 3xy(x-y-z)
= (x+y+z)[(x+y+z)² - 3z(x+y) - 3xy]
= (x+y+z)(x² + y² + z² + 2xy + 2xz + 2yz - 3xz - 3yz - 3xy)
= (x+y+z)(x² + y² + z² - xy - xz - yz).
\(P=x^3\left(z-y^2\right)+y^3\left(x-z^2\right)+z^3\left(y-x^2\right)+xyz\left(xyz-1\right)\)
\(P=\left(-x^3\left(y^2-z\right)\right)+xy^3-y^3z^2+yz^3-x^2z^3+x^2y^2z^2-xyz\)
\(P=\left(-x^3\left(y^2-z\right)\right)+\left(xy^3-xyz\right)-\left(y^3z^2-yz^3\right)+\left(x^2y^2z^2-x^2z^3\right)\)
\(P=\left(-x^3\left(y^2-z\right)\right)+\left(xy\left(y^2-z\right)\right)-\left(yz^2\left(y^2-z\right)\right)+\left(x^2z^2\left(y^2-z\right)\right)\)
\(P=\left(-x^3+xy-yz^2+x^2z^2\right)\left(y^2-z\right)\)
\(P=\left(\left(x^2z^2-x^3\right)-\left(yz^2-xy\right)\right)\left(y^2-z\right)\)
\(P=\left(x^2\left(z^2-x\right)-y\left(z^2-x\right)\right)\left(y^2-z\right)\)
\(P=\left(\left(x^2-y\right)\left(z^2-x\right)\right)\left(y^2-z\right)\)
\(P=\left(a.c\right).b\)
\(P=a.b.c\)
Vậy giá trị của P không phụ thuộc vào biến x;y;z (điều cần chứng minh)
\(15\left(2a^2-1\right)+5\left(3-\frac{1}{5a}-6a^2\right)\)
\(=30a^2-15+15-\frac{1}{a}-30a^2\)
\(=-\frac{1}{a}\)
tại \(a=2017\)=> M= \(\frac{-1}{a}=\frac{-1}{2017}\)
\(\left(x-y\right)\left(x^2+xy+y^2\right)+y^3\)
\(=x^3-y^3+y^3\)
\(=x^3\)
ại \(x=2\)=> N= \(x^3=2^3=8\)
b) Từ: x + y + z = 0
=> x + y = -z
<=> (x + y)^3 = (-z)^3
<=> x^3 + 3x^2y + 3xy^2 + y^3 = -z^3
<=> x^3 + y^3 + z^3 = -3x^2y - 3xy^2
<=> x^3 + y^3 + z^3 = -3xy(x+y)
<=> x^3 + y^3 + z^3 = -3xy(-z)
<=> x^3 + y^3 + z^3 = 3xyz
Chúc bạn học giỏi!! ^^
ok mk nhé!!! 657758768768768769675685876897696989789787809836366543453456567
tham khảo
https://olm.vn/hoi-dap/detail/6401290031.html
Gửi riêng
Ta có:
P=x3(z−y2)+y3(x−z2)+z3(y−x2)+xyz(xyz−1)P=x3(z−y2)+y3(x−z2)+z3(y−x2)+xyz(xyz−1)
=x3(z−y2)+xy3+yz3+x2y2z2−y3z2−z3x2−xyz=x3(z−y2)+xy3+yz3+x2y2z2−y3z2−z3x2−xyz
=x3(z−y2)+(xy3−xyz)+(yz3−y3z2)+(x2y2z2−z3x2)=x3(z−y2)+(xy3−xyz)+(yz3−y3z2)+(x2y2z2−z3x2)
=x3(z−y2)+xy(y2−z)+yz2(z−y2)+x2z2(y2−z)=x3(z−y2)+xy(y2−z)+yz2(z−y2)+x2z2(y2−z)
=(y2−z)(−x3+xy−yz2+x2z2)=(y2−z)(−x3+xy−yz2+x2z2)
=(y2−z)[x2(z2−x)−y(z2−x)]=(y2−z)[x2(z2−x)−y(z2−x)]
=(y2−z)(z2−x)(x2−y)=bca
\(\dfrac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2}\)
\(\Rightarrow\dfrac{\left(x+y\right)^3-3x^2y-3xy^2-3xyz+z^3}{x^2-2xy+y^2+y^2-2yz+z^2+x^2-2xz+z^2}\)
\(\Rightarrow\dfrac{\left[\left(x+y\right)^3+z^3\right]-3xy\left(x+y+z\right)}{2x^2+2y^2+2z^2-2xy-2yz-2xz}\)
\(\Rightarrow\dfrac{\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)}{2\left(x^2+y^2+z^2-xy-yz-xz\right)}\)
\(\Rightarrow\dfrac{\left(x+y+z\right)\left(x^2+2xy+z^2-xz-yz+z^2-3xy\right)}{2\left(x^2+y^2+z^2-xy-yz-xz\right)}\)
\(\Rightarrow\dfrac{\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)}{2\left(x^2+y^2+z^2-xy-yz-xz\right)}\)
\(\Rightarrow\dfrac{x+y+z}{2}\)
\(\Rightarrow\dfrac{1}{2}\left(x+y+z\right)\)
cảm ơn bạn rất nhiều nha :))