Chapter 4, Problem 11PS

(a)

To determine

The three real zeroes of the polynomial function $f$ using the given table,

$\begin{array}{|cc|}\hline \text{Interval}& \text{Value of }f\left(x\right)\\ \left(-\infty ,-2\right)& \text{Positive}\\ \left(-2,1\right)& \text{Negative}\\ \left(1,4\right)& \text{Negative}\\ \left(4,\infty \right)& \text{Positive}\\ \hline\end{array}$

(b)

To determine

The behaviour of graph of the polynomial function $f$ at $x=1$ with the help of the given table,

$\begin{array}{|cc|}\hline \text{Interval}& \text{Value of }f\left(x\right)\\ \left(-\infty ,-2\right)& \text{Positive}\\ \left(-2,1\right)& \text{Negative}\\ \left(1,4\right)& \text{Negative}\\ \left(4,\infty \right)& \text{Positive}\\ \hline\end{array}$

(c)

To determine

The least possible degree of the polynomial function $f$ with the help of the given table,

$\begin{array}{|cc|}\hline \text{Interval}& \text{Value of }f\left(x\right)\\ \left(-\infty ,-2\right)& \text{Positive}\\ \left(-2,1\right)& \text{Negative}\\ \left(1,4\right)& \text{Negative}\\ \left(4,\infty \right)& \text{Positive}\\ \hline\end{array}$

And, to determine that the degree of $f$ can ever be odd.

(d)

To determine

Whether the leading coefficient of $f$ is positive or negative.

$\begin{array}{|cc|}\hline \text{Interval}& \text{Value of }f\left(x\right)\\ \left(-\infty ,-2\right)& \text{Positive}\\ \left(-2,1\right)& \text{Negative}\\ \left(1,4\right)& \text{Negative}\\ \left(4,\infty \right)& \text{Positive}\\ \hline\end{array}$

(e)

To determine

The equation for the polynomial function $f$ with the help of the given table,

$\begin{array}{|cc|}\hline \text{Interval}& \text{Value of }f\left(x\right)\\ \left(-\infty ,-2\right)& \text{Positive}\\ \left(-2,1\right)& \text{Negative}\\ \left(1,4\right)& \text{Negative}\\ \left(4,\infty \right)& \text{Positive}\\ \hline\end{array}$

(f)

To determine

To graph: The graph of the function $f\left(x\right)=x^4-4x^3-3x^2+14x-8$.