Question

If the zeroes of the cubic polynomial f(x)=kx^3-8x^2+5 are alpha- beta, alpha,alpha+beta then find the value of k​

Answer 1

$f(x)=kx^3-8x^2+5=0\\ \\ a=k\ \ \ ; \ \ \ b=-8\ \ \ ; \ \ \ c=0\ \ \ ; \ \ \ d=5\\ \\ \\ \textsf{Let},\\ \\ \alpha-\beta=p\\ \\ \alpha=q\\ \\ \alpha+\beta=r$

So,

$p+q+r=-\dfrac{b}{a}\\ \\ \\ \alpha-\beta+\alpha+\alpha+\beta=-\dfrac{-8}{k}\\ \\ \\ 3\alpha=\dfrac{8}{k}\\ \\ \\ k=\dfrac{8}{3\alpha}\ \ \ \longrightarrow\ \ \ (1)$

$pq+qr+rp=\dfrac{c}{a}\\ \\ \\ (\alpha-\beta)\alpha + \alpha(\alpha+\beta)+(\alpha+\beta)(\alpha-\beta)=\dfrac{0}{k}\\ \\ \\ \alpha^2-\alpha\beta+\alpha^2+\alpha\beta+\alpha^2-\beta^2=0\\ \\ \\ 3\alpha^2-\beta^2=0\\ \\ \\ 3\alpha^2=\beta^2\\ \\ \\ \alpha\sqrt{3}=\pm\beta\ \ \ \longrightarrow\ \ \ (2)$

$pqr=-\dfrac{d}{a}\\ \\ \\ (\alpha-\beta)\alpha(\alpha+\beta)=-\dfrac{5}{k}\\ \\ \\ \alpha^3-\alpha\beta^2=-\dfrac{5}{k}\\ \\ \\ \alpha^3-\alpha(\pm\alpha\sqrt{3})^2=-\dfrac{5}{k}\ \ \ \ \ [\textsf{From}\ (2)]\\ \\ \\ \alpha^3\mp3\alpha^3=-\dfrac{5}{k}\\ \\ \\ -2\alpha^3=-\dfrac{5}{k}\ \ \ \ \ OR\ \ \ \ \ 4\alpha^3=-\dfrac{5}{k}\\ \\ \\ 2\alpha^3=\dfrac{5}{k}\ \ \ \ \ OR\ \ \ \ \ -4\alpha^3=\dfrac{5}{k}\\ \\ \\ k=\dfrac{5}{2\alpha^3}\ \ \ \ \ OR\ \ \ \ \ k=-\dfrac{5}{4\alpha^3}\ \ \ \longrightarrow\ \ \ (3)$

From (1) and (3),

$\dfrac{8}{3\alpha}=\dfrac{5}{2\alpha^3}\ \ \ \ \ OR\ \ \ \ \ \dfrac{8}{3\alpha}=-\dfrac{5}{4\alpha^3}\\ \\ \\ 16\alpha^3=15\alpha\ \ \ \ \ OR\ \ \ \ \ 32\alpha^3=-15\alpha\\ \\ \\ 16\alpha^2=15\ \ \ \ \ OR\ \ \ \ \ 32\alpha^2=-15\\ \\ \\ \alpha^2=\dfrac{15}{16}\ \ \ \ \ OR\ \ \ \ \ \alpha^2=-\dfrac{15}{32}\\ \\ \\ \alpha=\pm \dfrac{\sqrt{15}}{4}\ \ \ \ \ OR\ \ \ \ \ \alpha=\mp\dfrac{\sqrt{15}}{4\sqrt2}$

From (1),

$k=\dfrac{8}{3\alpha}\\ \\ \\ k=8\div \left(\pm \dfrac{3\sqrt{15}}{4}\right)\ \ \ \ \ OR\ \ \ \ \ k=8\div \left(\mp\dfrac{3\sqrt{15}}{4\sqrt2}\right)\\ \\ \\ k=8\times \left(\pm \dfrac{4}{3\sqrt{15}}\right)\ \ \ \ \ OR\ \ \ \ \ k=8\times \left(\mp\dfrac{4\sqrt2}{3\sqrt{15}}\right)\\ \\ \\ k=\pm \dfrac{32}{3\sqrt{15}}\ \ \ \ \ OR\ \ \ \ \ k=\mp\dfrac{32}{3}\sqrt{\dfrac{2}{15}}$