Question - Algebraic Expression Transformation

令 \( y + \frac{1}{y} = b \)

要表达 \( y^{2} + \frac{1}{y^{2}} \)，首先将原方程两边平方:

\( (y + \frac{1}{y})^{2} = b^{2} \)

展开左边，应用平方差公式得到:

\( y^{2} + 2 \cdot y \cdot \frac{1}{y} + \frac{1}{y^{2}} = b^{2} \)

简化得到:

\( y^{2} + 2 + \frac{1}{y^{2}} = b^{2} \)

现在将方程两边同时减去2:

\( y^{2} + \frac{1}{y^{2}} = b^{2} - 2 \)

因此，\( y^{2} + \frac{1}{y^{2}} \) 表达为 \( b \) 的项是:

\( y^{2} + \frac{1}{y^{2}} = b^{2} - 2 \)