Paragraph: Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots…
Paragraph:
Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.Question:
The real number $s$ lies in the interval
$\left(-\frac{1}{4}, 0\right)$
$\left(-11,-\frac{3}{4}\right)$
$\left(-\frac{3}{4},-\frac{1}{2}\right)$
$\left(0, \frac{1}{4}\right)$
Solution
Given, $f(x)=4 x^3+3 x^2+2 x+1$
$
\begin{aligned}
& f^{\prime}(x)=2\left(6 x^2+3 x+1\right) \\
& D=9-24 < 0
\end{aligned}
$
Hence, $f(x)=0$ has only one real root.
$
\begin{aligned}
f\left(-\frac{1}{2}\right) & =1-1+\frac{3}{4}-\frac{4}{8}>0 \\
f\left(-\frac{3}{4}\right) & =1-\frac{6}{4}+\frac{27}{16}-\frac{108}{64} \\
& =\frac{64-96+108-108}{64} < 0
\end{aligned}
$
$f(x)$ changes its sign in $\left(-\frac{3}{4}, \frac{-1}{2}\right)$, hence $f(x)=0$ has a root in $\left(\frac{-3}{4}, \frac{-1}{2}\right)$