Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 343

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\[1)\ (y - z)(y + z)^{2} +\]

\[+ (z - x)(z + x)^{2} +\]

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

\[= (y - z)\left( y^{2} + 2yz + z^{2} \right) +\]

\[+ (z - x)\left( z^{2} + 2zx + x^{2} \right) +\]

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

\[= y^{3} + 2y^{2}z + yz^{2} - y^{2}z -\]

\[- 2yz^{2} - z^{3} + 2xz^{2} + x^{2}z -\]

\[- z^{2}x - 2x^{2}z - x^{3} +\]

\[+ x^{3} + 2x^{2}y + xy^{2} - x^{2}y -\]

\[- 2xy^{2} - y^{3} =\]

\[= y^{2}z - yz^{2} + xz^{2} - x^{2}z +\]

\[+ x^{2}y - xy^{2} =\]

\[= z\left( y^{2} - x^{2} \right) - z^{2}(y - x) -\]

\[- xy(y - x) =\]

\[= (y - x)\left( z(y + x) - z^{2} - xy \right) =\]

\[= (y - x)\left( zy + zx - z^{2} - xy \right) =\]

\[= (y - x)\left( z(y - z) - x(y - z) \right) =\]

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

\[2)\ x^{6} - y^{6} + \left( x^{4} + x^{2}y^{2} + y^{4} \right) =\]

\[= \left( \left( x^{2} \right)^{3} - \left( y^{2} \right)^{3} \right) +\]

\[+ \left( x^{4} + x^{2}y^{2} + y^{4} \right) =\]

\[= (x^{2} - y^{2})(\left( x^{4} + x^{2}y^{2} + y^{4} \right) +\]

\[+ \left( x^{4} + x^{2}y^{2} + y^{4} \right) =\]

\[= \left( x^{4} + x^{2}y^{2} + y^{4} \right)\left( x^{2} - y^{2} + 1 \right) =\]