Решебник по алгебре 7 класс Мерзляк Задание 1202

\[\boxed{\text{1202\ (1202).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[x + y = a,\ \ xy = b\]

\[1)\ x^{2} + y^{2} = a^{2} - 2b\]

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

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

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

\[2)\ x^{3} + y^{3} = a^{3} - 3ab\]

\[a^{3} - 3ab =\]

\[= (x + y)^{3} - 3(x + y)xy =\]

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

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

\[3)\ x^{4} + y^{4} = a^{4} - 4a^{2}b + 2b^{2}\]

\[так\ как\ x^{2} + y^{2} = a^{2} - 2b,\]

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

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

\[x^{4} + 2x^{2}y^{2} + y^{4} =\]

\[= a^{4} - 4a^{2}b + 2b^{2}\]

\[x^{4} + y^{4} =\]

\[= a^{4} - 4a^{2}b + 4b^{2} - 2b^{2}\]

\[x^{4} + y^{4} = a^{4} - 4a^{2}b + 2b^{2}\ \]