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(A) Given $\alpha + \beta = p$ and $\alpha\beta = q$.Let the roots of the required quadratic equation be $A$ and $B$.$A = (\alpha^2 - \beta^2)(\alpha^

Vedclass · Accepted Answer

$x^2 - Sx + P = 0$

Vedclass · Answer

(A) Given $\alpha + \beta = p$ and $\alpha\beta = q$.Let the roots of the required quadratic equation be $A$ and $B$.$A = (\alpha^2 - \beta^2)(\alpha^3 - \beta^3) = (\alpha - \beta)(\alpha + \beta)(\alpha - \beta)(\alpha^2 + \alpha\beta + \beta^2) = (\alpha - \beta)^2(\alpha + \beta)(\alpha^2 + \alpha\beta + \beta^2)$.Since $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = p^2 - 4q$ and $\alpha^2 + \alpha\beta + \beta^2 = (\alpha + \beta)^2 - \alpha\beta = p^2 - q$,we have $A = (p^2 - 4q)p(p^2 - q) = p[p^4 - 5p^2q + 4q^2]$.$B = \alpha^3\beta^2 + \alpha^2\beta^3 = \alpha^2\beta^2(\alpha + \beta) = q^2p$.Sum of roots $S = A + B = p[p^4 - 5p^2q + 4q^2] + pq^2 = p[p^4 - 5p^2q + 5q^2]$.Product of roots $P = A \cdot B = p[p^4 - 5p^2q + 4q^2] \cdot pq^2 = p^2q^2(p^4 - 5p^2q + 4q^2)$.The required quadratic equation is $x^2 - Sx + P = 0$.

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).

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