Question

If α, β are the roots of the equation \[x^2 + px + 1 = 0; \gamma, \delta\] the roots of the equation \[x^2 + qx + 1 = 0, \text { then } (\alpha - \gamma)(\alpha + \delta)(\beta - \gamma)(\beta + \delta) =\]

Options

• \[q^2 - p^2\]

• \[p^2 - q^2\]

• \[p^2 + q^2\]

• none of these.

Answer 1

\[q^2 - p^2\]

Given: 

\[\alpha \text { and } \beta\] are the roots of the equation  \[x^2 + px + 1 = 0\].

Also,  

\[\gamma \text { and } \delta\] are the roots of the equation \[x^2 + qx + 1 = 0\].

Then, the sum and the product of the roots of the given equation are as follows:

\[\alpha + \beta = - \frac{p}{1} = - p\]

\[\alpha\beta = \frac{1}{1} = 1\]

\[\gamma + \delta = - \frac{q}{1} = - q\]

\[\gamma\delta = \frac{1}{1} = 1\]

\[\text { Moreover,} (\gamma + \delta )^2 = \gamma^2 + \delta^2 + 2\gamma\delta\]

\[ \Rightarrow \gamma^2 + \delta^2 = q^2 - 2\]

\[\therefore (\alpha - \gamma) (\alpha + \delta) (\beta - \gamma) (\beta + \delta) = (\alpha - \gamma) (\beta - \gamma) (\alpha + \delta) (\beta + \delta)\]

\[ = \left( \alpha\beta - \alpha\gamma - \beta\gamma + \gamma^2 \right)\left( \alpha\beta + \alpha\delta + \beta\delta + \delta^2 \right)\]

\[ = \left[ \alpha\beta - \gamma\left( \alpha + \beta \right) + \gamma^2 \right] \left[ \alpha\beta + \delta \left( \alpha + \beta \right) + \delta^2 \right]\]

\[ = (1 - \gamma( - p) + \gamma^2 ) (1 + \delta( - p) + \delta^2 )\]

\[ = (1 + \gamma p + \gamma^2 ) (1 - \delta p + \delta^2 )\]

\[ = 1 - p\delta + \delta^2 + p\gamma - p^2 \gamma\delta + p\gamma \delta^2 + \gamma^2 - p\delta \gamma^2 + \gamma^2 \delta^2 \]

\[ = 1 - p\delta + p\gamma + \delta^2 - p^2 \gamma\delta + p\gamma \delta^2 + \gamma^2 - p\delta \gamma^2 + \gamma^2 \delta^2 \]

\[ = 1 - p(\delta - \gamma) - p^2 \gamma\delta + p\gamma\delta (\delta - \gamma) + ( \gamma^2 + \delta^2 ) + 1\]

\[ = 1 - p^2 \gamma\delta + p\gamma\delta (\delta - \gamma) - p(\delta - \gamma) + ( \gamma^2 + \delta^2 ) + 1\]

\[ = 1 - p^2 + (\delta - \gamma) p (\gamma\delta - 1) + q^2 - 2 + 1\]

\[ = - p^2 + (\delta - \gamma) p (1 - 1) + q^2 \]

\[ = q^2 - p^2\]