Chapter 8.9, Problem 27WE

a.

To determine

To find: the location ofreal zero of polynomial function between consecutive tenths.

Answer to Problem 27WE

The real zero lie between $P\left(1.3\right)=-.203\text{ and }P\left(1.4\right)=0.544$

Explanation of Solution

Given information:

The polynomial is $P\left(x\right)=x^3+2x-5$ .

Calculation: to find the real zero, first find the polynomial function values,

$x$	$P\left(x\right)$
$0$	$-5$
$1$	$-2$
$2$	$7$

From the above table it can be observed that the polynomial zero can be lie in between $1-2$ , so now find the polynomial function values between $1-2$ ,

$x$	$P\left(x\right)$
$1.1$	$-1.469$
$1.3$	$-0.203$
$1.4$	$0.544$

Thus, the real zero of polynomial must be between in $1.3-1.4$ .

b.

To determine

To find: the zero to the nearest hundredth using linear interpolation.

Answer to Problem 27WE

The real zero is $r=1.32$ .

Explanation of Solution

Given information:

The polynomial is $P\left(x\right)=x^3+2x-5$ .

Calculation: to find the real zero, first find the polynomial function values,

$x$	$P\left(x\right)$
$0$	$-5$
$1$	$-2$
$2$	$7$

From the above table it can be observed that the polynomial zero can be lie in between $1-2$ , so now find the polynomial function values between $1-2$ ,

$x$	$P\left(x\right)$
$1.1$	$-1.469$
$1.3$	$-0.203$
$1.4$	$0.544$

Thus, the real zero of polynomial must be between in $1.3-1.4$ .

Let $r$ is the real zero of polynomial,

$x$	$P\left(x\right)$
$1.3$	$-0.203$
$r$	$0$
$1.4$	$0.544$

Now, to find the difference,

  $\begin{array}{l}\frac{c}{.1}=\frac{.203}{0.747}\\ \frac{c}{.1}=0.27\text{ [divide]}\\ c=0.027\text{ [multiply]}\end{array}$

Thus, the real zero of polynomial is,

  $\begin{array}{l}r=1.3+c\\ r=1.3+0.027\text{ [substitute]}\\ r=1.32\text{ [add]}\end{array}$