Chapter 8.8, Problem 29WE

To determine

To find : the least integral upper bound and greatest integral lower bound for the roots of the given equations.

Answer to Problem 29WE

Theleast upper bound for roots of $2x^3-5x^2-8x-9=0$ is $\text{4}$

The lower bound of the roots of the equation $2x^3-5x^2-8x-9=0$ is $-2$

Explanation of Solution

Given information :

The given equation:

  $2x^3-5x^2-8x-9=0$

Calculation :

As per the problem,

  $2x^3-5x^2-8x-9=0$

According to Exercise $27$ , $P(x)$ has the leading coefficient positive.

The given equation satisfies this condition.

Let $P(x)=2x^3-5x^2-8x-9$

  $P(x)=(x-m)Q(x)+P(m)$

Divide $P(x)=2x^3-5x^2-8x-9$ by $(x-m)$ using synthetic division for different values of m .

Take $c=m=1,2,3,\mathrm{...}$

Try until for a certain m all the coefficients of quotient is non-negative.

Now, for $c=m=4$

  $\begin{array}{l}\overline{)4}\text{ 2 }-5-8-9\\ \underset{\_}{\text{ 8 12 16}}\\ \text{ 2 3 4 | 7}\end{array}$

The quotient is $Q(x)=2x^2+3x+4$ and remainder is $7$ .

So, $P(x)=(x-4)(2x^2+3x+4)+7$

The quotient has all coefficients non-negative.

Now, $P(4)=2{(4)}^3-5{(4)}^2-8(4)-9=7>0$

  $P(3)=2{(3)}^3-5{(3)}^2-8(3)-9=-24<0$

So $P(x)$ has positive root between $3$ and $\text{4}$ .

So the least upper bound for roots of $2x^3-5x^2-8x-9=0$ is $\text{4}$ .

Similarly, to find the lower bound,

  $\begin{array}{l}P(-x)=-2x^3-5x^2\text{+ }8x-9\\ -P(-x)=2x^3+5x^2-8x+9\end{array}$

Divide $-P(-x)$ by $(x-m)$ using synthetic division for different values of m .

Take $c=m=1,2,3,\mathrm{...}$

Try until for a certain m all the coefficients of quotient is non-negative.

Now, for $c=m=2$

  $\begin{array}{l}\overline{)2}\text{2 }5-89\\ \underset{\_}{\text{ 4 18 20}}\\ \text{ 2 9 10 | 29}\end{array}$

Thus no root of $-P(-x)$ is greater than $2$ .

The positive roots of $-P(-x)$ is equal to the negative roots of $P(x)$ .

So, $P(x)=(x+2)(2x^2+9x+10)+29$

  $P(x)$ has no root less than $-2$ .

Thus, $-2$ is the lower bound of the roots of the equation $2x^3-5x^2-8x-9=0$