Question

If $\alpha ,\beta$ are the roots of $x^2-px+q=0$, then the product of the roots of the quadratic equation whose roots are ${\alpha }^2-{\beta }^2$ and ${\alpha }^3-{\beta }^3$ is

• A. $p(p^2-q{)}^2$
• B. $p(p^2-q)(p^2-4q)$
• C. $p(p^2-4q)(p^2+q)$
• D. none of these

Answer 1

Answer: (B)

$p(p^2-q)(p^2-4q)$

We need to find the product of (${\alpha }^2-{\beta }^2$) and (${\alpha }^3-{\beta }^3$)
So (${\alpha }^2-{\beta }^2$) * (${\alpha }^3-{\beta }^3$)$=$($\alpha -\beta {)}^2$* ($\alpha$+$\beta$)[($\alpha +\beta {)}^2-\alpha \beta$], Substituting the value $\alpha +\beta =p$ and $\alpha -\beta =q$and $(\alpha -\beta {)}^2=p^2-4q$
Now, $(\alpha +\beta )(\alpha -\beta {)}^2)\left[\right(\alpha +\beta {)}^2-\alpha \beta ]=p(p^2-q\left)\right(p^2-4q)$
Hence,option 'B' is correct.