Chapter 8.9, Problem 30WE

a.

To determine

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

Answer to Problem 30WE

The real zero lie between $P\left(-1.7\right)=-.313\text{ and }P\left(-1.6\right)=0.704$

Explanation of Solution

Given information:

The polynomial is $P\left(x\right)=x^3+2x+8$ .

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

$x$	$P\left(x\right)$
$0$	$8$
$-1$	$5$
$-2$	$-4$

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

$x$	$P\left(x\right)$
$-1.8$	$-1.432$
$-1.7$	$-0.313$
$-1.6$	$0.704$

Thus, the real zero of polynomial must be between in $\left(-1.8\right)-\left(-1.6\right)$ .

b.

To determine

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

Answer to Problem 30WE

The real zero is $r=-1.39$ .

Explanation of Solution

Given information:

The polynomial is $P\left(x\right)=x^3+2x+8$ .

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

$x$	$P\left(x\right)$
$0$	$8$
$-1$	$5$
$-2$	$-4$

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

$x$	$P\left(x\right)$
$-1.8$	$-1.432$
$-1.7$	$-0.313$
$-1.6$	$0.704$

Thus, the real zero of polynomial must be between in $\left(-1.8\right)-\left(-1.6\right)$ .

Let $r$ is the real zero of polynomial,

$x$	$P\left(x\right)$
$-1.7$	$-0.313$
$r$	$0$
$-1.6$	$0.704$

Now, to find the difference,

  $\begin{array}{l}\frac{c}{.1}=\frac{.313}{1.017}\\ \frac{c}{.1}=0.30\text{ [divide]}\\ c=0.030\text{ [multiply]}\end{array}$

Thus, the real zero of polynomial is,

  $\begin{array}{l}r=-1.7+c\\ r=-1.7+0.030\text{ [substitute]}\\ r=-1.39\text{ [add]}\end{array}$