Question

Find the integral part of the greatest root of equation $x^3-10x^2-11x-100=0$.

Answer 1

Given equation is

$x^3-10x^2-11x-100=0$

Let $f\left(x\right)=x^3-10x^2-11x-100$

$\Rightarrow f^{\prime }\left(x\right)=3x^2-20x-11$

For $3x^2-20x-11=0,$

we have$x=\frac{20\pm \sqrt{400+132}}{6}=\frac{10\pm \sqrt{133}}{3}$

Hence, graph of $y=f\left(x\right)$ is plotted

Now, $(10+\sqrt{133})/3\cong 7.16$

$f\left(8\right)=8^3-10(8{)}^2-11(8)-100<0$

$f\left(9\right)=9^3-10(9{)}^2-11(9)-100<0$

$f\left(10\right)={10}^3-10(10{)}^2-11(10)-100<0$

$f\left(11\right)={11}^3-10(11{)}^2-11(11)-100<0$

$f\left(12\right)={12}^3-10(12{)}^2-11(12)-100>0$

$\Rightarrow y\in (11,12)$

$\Rightarrow \left[y\right]=11$