Bài 8:
\(M=\left(a+b\right)^3-3ab\left(a+b\right)+3ab\left[\left(a+b\right)^2-2ab\right]+6a^2b^2\)
\(=1-3ab+3ab-6a^2b^2+6a^2b^2=1\)
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Bài 7:
\(ax+by+cz=0\)
\(\Rightarrow\left(ax+by+cz\right)^2=0\)
\(\Rightarrow a^2x^2+b^2y^2+c^2z^2+2axby+2axcz+2bycz=0\)
\(\Rightarrow a^2x^2+b^2y^2+c^2z^2=-2axby-2axcz-2bycz\).
\(A=\dfrac{bc\left(y-z\right)^2+ca\left(z-x\right)^2+ab\left(x-y\right)^2}{ax^2+by^2+cz^2}\)
\(=\dfrac{bc\left(y^2-2yz+z^2\right)+ca\left(z^2-2zx+x^2\right)+ab\left(x^2-2xy+y^2\right)}{ax^2+by^2+cz^2}\)
\(=\dfrac{bcy^2-2bcyz+bcz^2+caz^2-2cazx+cax^2+abx^2-2abxy+aby^2}{ax^2+by^2+cz^2}\)
\(=\dfrac{bcy^2+bcz^2+caz^2+cax^2+abx^2+aby^2-2bcyz-2cazx-2abxy}{ax^2+by^2+cz^2}\)
\(=\dfrac{bcy^2+bcz^2+caz^2+cax^2+abx^2+aby^2+a^2x^2+b^2y^2+c^2z^2}{ax^2+by^2+cz^2}\)
\(=\dfrac{ax^2\left(c+b+a\right)+by^2\left(c+a+b\right)+cz^2\left(b+a+c\right)}{ax^2+by^2+cz^2}\)
\(=\dfrac{\left(a+b+c\right)\left(ax^2+by^2+cz^2\right)}{ax^2+by^2+cz^2}=a+b+c\)
