Question

Consider the polynomial $f\left(x\right)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f\left(x\right)$ and let $t=|s|$.

The real number $s$ lies in the interval

• A. $\left(\frac{-1}{4},0\right)$
• B. $(-11,\frac{-3}{4})$
• C. $\left(\frac{-3}{4},\frac{-1}{2}\right)$
• D. $(0,\frac{1}{4})$

Answer 1

The correct option is C $\left(\frac{-3}{4},\frac{-1}{2}\right)$

$f\left(x\right)=1+2x+3x^2+4x^3$
$f^{\prime }\left(x\right)=2+6x+12x^2$
$f^{\prime }\left(x\right)=2(1+3x+6x^2)$
$D\Rightarrow b^2-4ac$ (ignoring the constant multiplier $2$).
$D\Rightarrow 9-4\times 6$
$\Rightarrow -15$
This means it is always increasing function. We can also conclude that f(x) has only one real root.

Now, check the interval mentioned in the options $a<f\left(x\right)<b$
$f\left(a\right)f\left(b\right)<0\Rightarrow \text{root lies in}(\frac{-3}{4},\frac{-1}{2})$