Find interval and root of polynomial with absolute error less than 1/8

[songoku]

Homework Statement

Find an interval of length $\frac{1}{4}$ which contains root of $x^3+2x+1=0$. Then using that interval, give approximate value of the root with absolute error less than $\frac{1}{8}$

Relevant Equations

Intermediate Value Theorem (IVT)

Limit (maybe)

By IVT and trial and error, I get the interval to be $(-\frac{1}{2},-\frac{1}{4})$

I don't know how to do the next part.

Let the actual root of the polynomial be $x_{0}$ and the approximate value is $p$, we have $|p-x_{0}|<\frac{1}{8}$

I am not sure how to continue.

Thanks

 

[anuttarasammyak]

https://www.wolframalpha.com/input?i=plot+y=x^3+2x+1&lang=en

$$f(x)=x^3+2x+1$$
Say [a,b] is the range we choose, linear line between, i.e.,
$$y-f(a)=\frac{f(b)-f(a)}{b-a}(x-a)$$
cross the x axis at
$$x=-\frac{(b-a)f(a)}{f(b)-f(a)}+a$$
This is a candidate of approximate solution which should be investigated to be within the cited error.

[EDIT]
f(a)<0, f(b)>0
If f($\frac{a+b}{2}$)<0 we can halve the size of the section [$\frac{a+b}{2}$,b].
If f($\frac{a+b}{2}$)>0 we can halve the size of the section [a,$\frac{a+b}{2}$].

 

[songoku]

https://www.wolframalpha.com/input?i=plot+y=x^3+2x+1&lang=en
View attachment 352641

$$f(x)=x^3+2x+1$$
Say [a,b] is the range we choose, linear line between, i.e.,
$$y-f(a)=\frac{f(b)-f(a)}{b-a}(x-a)$$
cross the x axis at
$$x=-\frac{(b-a)f(a)}{f(b)-f(a)}+a$$
This is a candidate of approximate solution which should be investigated to be within the cited error.

[EDIT]
f(a)<0, f(b)>0
If f($\frac{a+b}{2}$)<0 we can halve the size of the section [$\frac{a+b}{2}$,b].
If f($\frac{a+b}{2}$)>0 we can halve the size of the section [a,$\frac{a+b}{2}$].

So the idea is to use linearization and then change the interval based on the value of $x$ obtained from linearization until the interval length is < $\frac{1}{8}$?

Thanks

 

[anuttarasammyak]

f(a)<0, f(b)>0
If f((a+b)/2)<0 we can halve the size of the section [(a+b)/2,b].
If f((a+b)/2)>0 we can halve the size of the section [a,(a+b)/2].

Please forget the part before [EDIT]. Bisection method https://en.wikipedia.org/wiki/Bisection_method applies here.　We can get better convergence by Newton method.

 

[songoku]

Thank you very much for the help and explanation anuttarasammyak