Question
Write the following squares of binomials as trinomials:
$\big(\frac{\text{x}}{\text{y}}−\frac{\text{y}}{\text{x}})^2$
$\big(\frac{\text{x}}{\text{y}}−\frac{\text{y}}{\text{x}})^2$
✓
Answer
We will use the identities $(a+b)^2=a^2+2 a b+b^2$ and $(a-b)^2=a^2-2 a b+b^2$ to convert the squares of binomials as trinomials.
$\big(\frac{\text{x}}{\text{y}}−\frac{\text{y}}{\text{x}})^2$
$=\big(\frac{\text{x}}{\text{y}})^2−2\big(\frac{\text{x}}{\text{y}}\big)\big(\frac{\text{y}}{\text{x}}\big)+\big(\frac{\text{y}}{\text{x}})^2$
$=\frac{\text{x}^2}{\text{y}^2}−2+\frac{\text{y}^2}{\text{x}^2}$
$\big(\frac{\text{x}}{\text{y}}−\frac{\text{y}}{\text{x}})^2$
$=\big(\frac{\text{x}}{\text{y}})^2−2\big(\frac{\text{x}}{\text{y}}\big)\big(\frac{\text{y}}{\text{x}}\big)+\big(\frac{\text{y}}{\text{x}})^2$
$=\frac{\text{x}^2}{\text{y}^2}−2+\frac{\text{y}^2}{\text{x}^2}$
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