Chapter 8.8, Problem 15WE

To determine

To calculate : the real root of the given equation $P(x)=0$ and approximate it to the nearest tenth.

Answer to Problem 15WE

The real root of the given equation is $2.1$

Explanation of Solution

Given information :

The polynomial equation:

  $P(x)=x^3-x^2-x-3$

Formula used :

Intermediate-Value Theorem:

If P is a polynomial function with real coefficients, and m is any number between P (a) and P ( b ), then there is at least one number c between a and b for which P ( c ) = m .

Calculation :

As per the given problem,

Consider the given equation:

  $P(x)=x^3-x^2-x-3$

By Descartes’s rule $P(x)=0$ has no negative roots. Make a table using non-negative values of x.

The table of values at the left below shows that $P(2)<0$ and $P(3)>0$ . Therefore, for some r between $2$ and $3$ , $P(r)=0$ .

  

The table of values at the right above, shows that r is between $2.1$ and $2.2$ . Since $P(2.1)$ is closer to $0$ than $P(2.2)$ , it can be assumed that r is closer to $2.1$ . Therefore, to the nearest tenth, the real zero of P is $2.1$