Question

Rút gọn phân thức:

a, \(\dfrac{x^3-x^2y+xy^2-y^3}{x^2y+xy^2-x^3-y^3}\)

b, \(\dfrac{a^3+b^3+c^3-3abc}{\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3}\) biết a+b+c = 3

Answer 1

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Answer 2

a/ \(\dfrac{x^3-x^2y+xy^2-y^3}{x^2y+xy^2-x^3-y^3}\)

\(=\dfrac{x^2\left(x-y\right)+y^2\left(x-y\right)}{x^2\left(y-x\right)+y^2\left(x-y\right)}\)

\(=\dfrac{\left(x-y\right)\left(x^2+y^2\right)}{\left(x-y\right)\left(y-x\right)\left(y+x\right)}\)

\(=\dfrac{x^2+y^2}{\left(y-x\right)\left(y+x\right)}\)

Answer 3

b: \(=\dfrac{\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3bac}{\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3}\)

\(=\dfrac{\left(a+b+c\right)\left(a^2+2ba+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)}{\left(a-b+b-c\right)^3-3\left(a-b\right)\left(b-c\right)\left(a-b+b-c\right)+\left(c-a\right)^3}\)

\(=\dfrac{3\left(a^2+b^2+c^2-ab-bc-ac\right)}{-3\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)

\(=\dfrac{a^2+b^2+c^2-ab-bc-ac}{\left(b-a\right)\left(b-c\right)\left(a-c\right)}\)

a: \(=\dfrac{x^2\left(x-y\right)+y^2\left(x-y\right)}{xy\left(x+y\right)-\left(x+y\right)\left(x^2-xy+y^2\right)}\)

\(=\dfrac{\left(x-y\right)\left(x^2+y^2\right)}{\left(x+y\right)\left(xy-x^2+xy-y^2\right)}\)

\(=\dfrac{\left(x-y\right)\left(x^2+y^2\right)}{-\left(x+y\right)\left(x-y\right)^2}=\dfrac{-\left(x^2+y^2\right)}{x^2-y^2}\)