Chapter 8.8, Problem 19WE

To determine

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

Answer to Problem 19WE

The roots of the equation are $1.5\text{ and }2.8$

Explanation of Solution

Given information :

The polynomial equation:

  $P(x)=x^4-3x^3+5$

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:

  $P(x)=x^4-3x^3+5$

By Descartes’s rule $P(x)=0$ has only two positive roots. Make a table using non-negative values of x.

The table of values below shows that $P(1)>0$ , $P(2)<0$ and $P(3)>0$ . Therefore, the positive roots lie between $1$ and $2$ , and between $2$ and $3$ .

  

The table of values at the left below, shows that one of the positive roots is between $1.4$ and $1.5$ Since $P(1.5)$ is closer to $0$ than $P(1.4)$ , it can be assumed that root is closer to $1.5$ . Therefore, to the nearest tenth, one of the positive roots of P is $1.5$

The table of values at the right below, shows that the other positive root is between $2.7$ and $2.8$ Since $P(2.8)$ is closer to $0$ than $P(2.7)$ , it can be assumed that root is closer to $2.8$ . Therefore, to the nearest tenth, the other negative root of P is $2.8$

  

The roots of P are $1.5\text{ and }2.8$