* Rút gọn phân thức:
a. \(\dfrac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ac}\)
b. \(\dfrac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
d. \(\dfrac{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}{a^4\left(b^2-c^2\right)+b^4\left(c^2-a^2\right)+c^4\left(a^2-b^2\right)}\)
e. \(\dfrac{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}{ab^2-ac^2-b^3+bc^2}\)
@Lê Gia Bảo @Lê Thị Nơ
Cứu trẫm. :3
a. \(\dfrac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ac}=\dfrac{\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc}{a^2+b^2+c^2-ab-bc-ac}\)
\(=\dfrac{\left(a+b+c\right)\left[\left(a+b\right)^2-\left(a+b\right)c+c^2\right]-3ab\left(a+b+c\right)}{a^2+b^2+c^2-ab-bc-ac}\)
\(=\dfrac{\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2-3ab\right)}{a^2+b^2+c^2-ab-bc-ac}\)
\(=\dfrac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)}{a^2+b^2+c^2-ab-bc-ac}=a+b+c\)
b. \(\dfrac{x^3-y^3+z^3+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}=\dfrac{\left(x-y\right)^3+z^3+3xy\left(x-y\right)+3xyz}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
\(=\dfrac{\left(x-y+z\right)\left[\left(x-y\right)^2-\left(x-y\right)z+z^2\right]+3xy\left(x-y+z\right)}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
\(=\dfrac{\left(x-y+z\right)\left(x^2-2xy+y^2-xz+yz+z^2+3xy\right)}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
\(=\dfrac{\left(x-y+z\right)\left(x^2+y^2+z^2+xy+yz-xz\right)}{\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2}\)
\(=\dfrac{2\left(x-y+z\right)\left(x^2+y^2+z^2+xy+yz-xz\right)}{2\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2\right]}\)
\(=\dfrac{\left(x+y-z\right)\left(2x^2+2y^2+2z^2+2xy+2yz-2zx\right)}{2\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2\right]}\)
\(=\dfrac{\left(x-y+z\right)\left[\left(x^2+2xy+y^2\right)+\left(y^2+2yz+z^2\right)+\left(z^2-2zx+x^2\right)\right]}{2\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2\right]}\)
\(=\dfrac{\left(x-y+z\right)\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2\right]}{2\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z-x\right)^2\right]}=\dfrac{x-y+z}{2}\)
