Question

f(x) 3x^3 + 10x^2 + 10x +7. find the interval of uni length which contains the root

Answer 1

this one actually factors, but you can try \(f(-1)\) and \(f(-2)\) and see that one is positive, the other negative

Answer 2

this is called the intermediate value theorem the results of the two y values given by satellite are between 0 \[f(-1)<f(0)<-2)\]
then there is a number c in the interval (-1,-2) such that \[f(c)=0\]

Answer 3

\[
3 x^3+10 x^2+10 x+7=(3 x+7)
\left(x^2+x+1\right)\\

x= -\frac 7 3
\]

Answer 4

\[
[-\frac 7 3 - \frac 1 2,-\frac 7 3 + \frac 1 2]=[-\frac {17}6,-\frac {11}6]
\]