Question

a, ß and y are zeroes of the polynomial x³ + px² + qx + 2 such that a. ß+1=0.
Find the value of 2p + q +5​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: \alpha, \beta, \gamma \: are \: zeroes \: of \: {x}^{3} + {px}^{2} + qx + 2$

and

$\rm :\longmapsto\: \alpha \beta + 1 = 0$

$\bf\implies \: \alpha \beta = - \: 1$

We know, that

$\rm :\longmapsto\: \alpha \beta \gamma = - \: \dfrac{constant \: term}{coefficient \: of \: {x}^{3} }$

Thus,

$\rm :\longmapsto\:( - 1) \gamma = - \: \dfrac{2}{1}$

$\bf\implies \: \gamma = 2$

Since,

$\rm :\longmapsto\: \gamma \: is \: zero \: of \: {x}^{3} + p {x}^{2} + qx + 2$

$\rm :\longmapsto\: 2 \: is \: zero \: of \: {x}^{3} + p {x}^{2} + qx + 2$

$\rm :\longmapsto\:{2}^{3} + p {(2)}^{2} + 2q + 2 = 0$

$\rm :\longmapsto\:8 + 4p + 2q + 2 = 0$

$\rm :\longmapsto \: 4p + 2q + 10 = 0$

$\rm :\longmapsto \: 2(2p + q + 5) = 0$

$\rm :\longmapsto \: 2p + q + 5= 0$

Thus,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf{ \: Value \: of \: 2p + q + 5 = 0}}}$

Additional Information :-

$\rm :\longmapsto\: \alpha, \beta, \gamma \: are \: zeroes \: of \: {ax}^{3} + {bx}^{2} + cx + d \: then \:$

$\boxed{ \sf \: \alpha + \beta + \gamma = - \: \frac{b}{a}}$

$\boxed{ \sf \: \alpha \beta + \beta \gamma + \gamma \alpha = \: \frac{c}{a}}$

$\boxed{ \sf \: \alpha \beta \gamma = - \: \frac{d}{a}}$