If alpha, beta are the roots of x^2+p x+q=0,t | vedclass

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(D) Given that $\alpha$ and $\beta$ are roots of the equation $x^2+p x+q=0$. From the relation between roots and coefficients,we have $\alpha+\beta =

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$(3 p q-p^3)$ and $(p^2-q)(p^2-3 q)$

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(D) Given that $\alpha$ and $\beta$ are roots of the equation $x^2+p x+q=0$. From the relation between roots and coefficients,we have $\alpha+\beta = -p$ and $\alpha \beta = q$. For $\alpha^3+\beta^3$: $\alpha^3+\beta^3 = (\alpha+\beta)(\alpha^2-\alpha \beta+\beta^2) = (\alpha+\beta)[(\alpha+\beta)^2-3 \alpha \beta] = (-p)[(-p)^2-3 q] = -p(p^2-3 q) = 3 p q-p^3$. For $\alpha^4+\alpha^2 \beta^2+\beta^4$: $\alpha^4+\alpha^2 \beta^2+\beta^4 = (\alpha^2+\beta^2)^2 - \alpha^2 \beta^2 = [(\alpha+\beta)^2-2 \alpha \beta]^2 - (\alpha \beta)^2 = [(-p)^2-2 q]^2 - q^2 = (p^2-2 q)^2 - q^2$. Using the identity $a^2-b^2 = (a-b)(a+b)$,we get $(p^2-2 q-q)(p^2-2 q+q) = (p^2-3 q)(p^2-q)$.

If $\alpha, \beta$ are the roots of $x^2+p x+q=0$,then the values of $\alpha^3+\beta^3$ and $\alpha^4+\alpha^2 \beta^2+\beta^4$ are respectively ...... and ......

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