Question

Rút gọn phân thức:

\(a,\dfrac{x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)}{x^2y-x^2z+y^2z-y^3}\)

\(b,\dfrac{x^5+x+1}{x^3+x^2+x}\)

Answer 1

Câu a:

Xét tử số:

\(x^2(y-z)+y^2(z-x)+z^2(x-y)\)

\(=x^2(y-z)-y^2[(y-z)+(x-y)]+z^2(x-y)\)

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

\(=(x^2-y^2)(y-z)-(y^2-z^2)(x-y)\)

\(=(x-y)(y-z)[(x+y)-(y+z)]=(x-y)(y-z)(x-z)\)

Xét mẫu số:

\(x^2y-x^2z+y^2z-y^3=x^2(y-z)-y^2(y-z)=(x^2-y^2)(y-z)\)

\(=(x-y)(x+y)(y-z)\)

Do đó:
\(\frac{x^2(y-z)+y^2(z-x)+z^2(x-y)}{x^2y-x^2z+y^2z-y^3}=\frac{(x-y)(y-z)(x-z)}{(x-y)(x+y)(y-z)}=\frac{x-z}{x+y}\)

Answer 2

Câu b:

Xét tử số:

\(x^5+x+1=x^5-x^2+x^2+x+1=x^2(x^3-1)+x^2+x+1\)

\(=x^2(x-1)(x^2+x+1)+(x^2+x+1)\)

\(=(x^2+x+1)(x^3-x^2+1)\)

Xét mẫu số:

\(x^3+x^2+x=x(x^2+x+1)\)

Do đó: \(\frac{x^5+x+1}{x^3+x^2+1}=\frac{(x^2+x+1)(x^3-x^2+1)}{x(x^2+x+1)}=\frac{x^3-x^2+1}{x}\)