Решебник по алгебре 8 класс Мерзляк Задание 467

\[\boxed{\text{467\ (467).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[Разложим\ первое\ равенство:\]

\[a^{2} \cdot (b + c) = b^{2} \cdot (c + a)\]

\[a^{2}b + a^{2}c = b^{2}c + ab^{2}\]

\[a^{2}b + a^{2}c - b^{2}c - ab^{2} = 0\]

\[\text{ab}(a - b) + c\left( a^{2} - b^{2} \right) = 0\]

\[\text{ab}(a - b) + c(a - b)(a + b) = 0\]

\[(a - b)\left( ab + c(a + b) \right) = 0\]

\[(a - b)(ab + ac + bc) = 0\]

\[ab + ac + bc = 0;\ \ \]

\[так\ как\ a - b \neq 0\ по\ условию\text{.\ }\]

\[Разложим\ второе\ равенство:\]

\[a^{2}(b + c) = c^{2}(a + b)\]

\[a^{2}b + a^{2}c = ac^{2} + bc^{2}\]

\[a^{2}b - bc^{2} + a^{2}c - ac^{2} = 0\]

\[b\left( a^{2} - c^{2} \right) + ac(a - c) = 0\]

\[b(a - c)(a + c) + ac(a - c) = 0\]

\[(a - c)\left( b(a + c) + ac \right) = 0\]

\[(a - c)(ab + bc + ac) = 0\]

\[ab + bc + ac = 0;\ \ \]

\[так\ как\ a - c \neq 0\ по\ условию.\]