CAT 2022 Slot 2 QA Question 16: Let $r$ and $c$ be real numbers. If $r$ and $-r$ are roots of $5x^3 + cx^2 - 10x + 9 = 0$, then $c$ equals

Question

Let $r$ and $c$ be real numbers. If $r$ and $-r$ are roots of $5x^3 + cx^2 - 10x + 9 = 0$, then $c$ equals

Accepted Answer

Let the third root be $p$. The roots are $r$, $-r$ and $p$.

For the cubic equation, the sum of roots is $-\dfrac{c}{5}$.

$r - r + p = - \dfrac{c}{5}$
$p = - \dfrac{c}{5}$

The sum of pairwise products is the coefficient of $x$ divided by the coefficient of $x^3$.

$r(-r) + rp + (-r)p = \dfrac{-10}

Let rrr and ccc be real numbers. If rrr and −r-r−r are roots of 5x3+cx2−10x+9=05x^3 + cx^2 - 10x + 9 = 05x3+cx2−10x+9=0, then ccc equals