If alpha and beta are the roots of x^{2}-px+1 | vedclass

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(A) Given that $\alpha$ and $\beta$ are the roots of $x^{2}-px+1=0$. By the relation between roots and coefficients, $\alpha+\beta=p$ and $\alpha\beta

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(A) Given that $\alpha$ and $\beta$ are the roots of $x^{2}-px+1=0$. By the relation between roots and coefficients, $\alpha+\beta=p$ and $\alpha\beta=1$. Also, $\gamma$ is a root of $x^{2}+px+1=0$, so $\gamma^{2}+p\gamma+1=0$, which implies $\gamma^{2}=-p\gamma-1$. Now, consider the expression $(\alpha+\gamma)(\beta+\gamma) = \alpha\beta + \gamma(\alpha+\beta) + \gamma^{2}$. Substituting the known values: $(\alpha+\gamma)(\beta+\gamma) = 1 + \gamma(p) + (-p\gamma-1)$. $= 1 + p\gamma - p\gamma - 1 = 0$.

If $\alpha$ and $\beta$ are the roots of $x^{2}-px+1=0$ and $\gamma$ is a root of $x^{2}+px+1=0,$ then $(\alpha+\gamma)(\beta+\gamma)$ is

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