Đặt \(A=x^{20}+x^{10}+1\)

\(x^{50}+x^{10}+1\)

\(=x^{50}-x^{20}+A\)

\(=x^{20}\left(x^{30}-1\right)+A\)

\(=x^{20}\left(x^{10}-1\right)A+A\)

\(=\left(x^{30}-x^{20}+1\right)A\)

mà \(\left(x^{30}-x^{20}+1\right)A⋮A\)

\(\Rightarrow\left(x^{50}+x^{10}+1\right)⋮\left(x^{20}+x^{10}+1\right)\)

Lời giải:

\(B=x(x^2+xy+y^2)-y(y^2+xy+y^2)\)

\(=(x-y)(x^2+xy+y^2)=x^3-y^3=10^3-(-1)^3=1000-(-1)=1001\)

\(C=x^4+10x^3+10x^2+10\)

\(=x^4+9x^3+x^3+9x^2+x^2+10\)

\(=x^3(x+9)+x^2(x+9)+x^2+10\)

\(=(x+9)(x^3+x^2)+x^2+10\)

\(=(-9+9)[(-9)^3+(-9)^2]+(-9)^2+10\)

\(=0+(-9)^2+10=91\)

Thay $x=-1$ vào biểu thức:

\(D=x^2(x+y)-xy(x-y)-x(y^2+1)\)

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

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

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

\(=(x+1)(y+1)=(-1+1)(y+1)=0\)

b: \(\Leftrightarrow\dfrac{7x+10}{x+1}\left(x^2-x-2-2x^2+3x+5\right)=0\)

\(\Leftrightarrow\left(7x+10\right)\left(-x^2+2x+3\right)=0\)

\(\Leftrightarrow\left(7x+10\right)\left(x^2-2x-3\right)=0\)

=>(7x+10)(x-3)=0

hay \(x\in\left\{-\dfrac{10}{7};3\right\}\)

d: \(\Leftrightarrow\dfrac{13}{2x^2+7x-6x-21}+\dfrac{1}{2x+7}-\dfrac{6}{\left(x-3\right)\left(x+3\right)}=0\)

\(\Leftrightarrow\dfrac{13}{\left(2x+7\right)\left(x-3\right)}+\dfrac{1}{\left(2x+7\right)}-\dfrac{6}{\left(x-3\right)\left(x+3\right)}=0\)

\(\Leftrightarrow26x+91+x^2-9-12x-14=0\)

\(\Leftrightarrow x^2+14x+68=0\)

hay \(x\in\varnothing\)

a) đặt \(\left(x^2+x\right)\)là \(y\)

ta có: \(3y^2-7y+4\)\(=0\)

<=>\(\left(3y-4\right)\left(y-1\right)=0\)

còn lại bạn tự xử nhé