Question

Show that $x^5+5x^4-20x^2-19x-2=0$ has a root between $2$ and $3$, and a root between $-4$ and $-5$.

Answer 1

Given equation, $x^5+5x^4-20x^2-19x-2=0$

Consider $f\left(x\right)=x^5+5x^4-20x^2-19x-2$

Then, $f\left(2\right)=-8$ and $f\left(3\right)=409$

Since the signs of $f\left(2\right)$ and $f\left(3\right)$ are opposite, $f\left(x\right)$ must cross x-axis atleast once in the interval $(2,3)$.

$\therefore f\left(x\right)=0$ must have a root between $2$ and $3$

Similarly, $f(-4)=10$ and $f(-5)=-407$.

Since the signs of $f(-4)$ and $f(-5)$ are different, $f\left(x\right)=0$ must have one root between $-4$ and $-5$.