Question

$\alpha ,\beta ,\gamma$ are zeros of cubic polynomial $x^3-12x^2+44x+c.$ If $\alpha ,\beta ,\gamma$ are in A.P. find the value of c

• A. $13$
• B. $-23$
• C. $-48$
• D. $19$

Answer 1

The correct option is C $-48$

$\alpha ,\beta$ and $\gamma$ are zeros of cubic polynomial and are in $AP$.

So, Let $\beta =a$ ; $\alpha =a-d$ & $\gamma =a+d$

Polynomial$=x^3-12x^2+44x+c$

Sum of roots$=\frac{-(-12)}{1}=12$

So,$a-d+a+a+d=12$

$3a=12a=4$

Sum of products of two consecutive roots$=44$.

$a(a-d)+a(a+d)+(a-d)(a+d)=44a^2-ad+a^2+ad+a^2-d^2=443a^2-d^2=443{\left(4\right)}^2-d^2=44d^2=48-44=4d=\pm 2$

So, $\alpha =a-d=\frac{4-2}{4+2}=\frac{2}{6}$

$\beta =4\beta =a+d=\frac{4+2}{4-2}=\frac{6}{2}$

So,Product $(-c)=2\times 4\times 6$

$=-48$