a: \(D=4\left(x-2\right)\left(x+1\right)+\left(2x-4\right)^2+\left(x+1\right)^2\)
\(=4\left(x^2-x-2\right)+4x^2-16x+16+x^2+2x+1\)
\(=4x^2-4x-8+5x^2-14x+17\)
\(=9x^2-18x+9=9\left(x^2-2x+1\right)=9\left(x-1\right)^2\)
Khi x=-1/2 thì \(D=9\cdot\left(-\dfrac{1}{2}-1\right)^2=9\cdot\left(-\dfrac{3}{2}\right)^2=9\cdot\dfrac{9}{4}=\dfrac{81}{4}\)
b: \(E=x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z+y^2z-xy^2+z^2\left(x-y\right)\)
\(=\left(x^2y-xy^2\right)-z\left(x^2-y^2\right)+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left(xy-zx-zy+z^2\right)\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)
\(=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
Khi x=6;y=5;z=4 thì \(E=\left(6-5\right)\cdot\left(6-4\right)\cdot\left(5-4\right)=2\)
c: x=9 nên x+1=10
\(F=x^{2017}-10x^{2016}+10x^{2015}-...-10x^2+10x-10\)
\(=x^{2017}-x^{2016}\left(x+1\right)+x^{2015}\left(x+1\right)-...-x^2\left(x+1\right)+x\left(x+1\right)-\left(x+1\right)\)
\(=x^{2017}-x^{2017}-x^{2016}+x^{2016}+...+x^2+x-x-1\)
=-1
d: x=100 nên x-1=99
\(D=99x^{100}+99x^{99}+99x^{98}+...+99x^2+99x+99\)
\(=99\left(x^{100}+x^{99}+...+x+1\right)\)
\(=\left(x-1\right)\left(x^{100}+x^{99}+...+x+1\right)\)
\(=x^{101}-1=100^{101}-1\)
b: \(D=y^3+4x^2y+4xy+8x^3+2xy^2\)
\(=\left(8x^3+y^3\right)+\left(4x^2y+4xy+2xy^2\right)\)
\(=\left(2x+y\right)^3-3\cdot2x\cdot y\left(2x+y\right)+2xy\left(2x+2+y\right)\)
\(=1^3-3\cdot6xy+2xy\cdot\left(1+2\right)=1\)
