Chapter 4.1, Problem 51AYU

In Problems 57-68, for each polynomial function:

(a) List each real zero and its multiplicity.

(b) Determine whether the graph crosses or touches the $x\text{-axis}$ at each $x\text{-intercept}$

(c) Determine the maximum number of turning points on the graph.

(d) Determine the end behavior; that is, find the power function that the graph of $f$ resembles for large values of $\left|x\right|$.

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

To determine

To find:

a. Each real zero and its multiplicity.

Answer to Problem 51AYU

a. $\text{Real zeros }=1,-1,-2,-2,-2$.

$\text{The multiplicity of the polynomial }=1\left(\text{Multiplicity }=1\right)$.

$\text{The multiplicity of the polynomial }=-1\left(\text{Multiplicity }=\text{ 1}\right)$.

$\text{The multiplicity of the polynomial }=-2\left(\text{Multiplicity }=\text{ 3}\right)$.

Explanation of Solution

Given:

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3=4\left(x^2+1\right)\left(x^3-6x^2+12x-8\right)$

$=4\left(x^5-6x^4+13x^3-14x^2+12x-8\right)$

a. The real zeros of the polynomial $=1,-1,-2,-2,-2$.

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

The multiplicity of the polynomial $=1\left(\text{Multiplicity }=1\right)$

The multiplicity of the polynomial $=-1\left(\text{Multiplicity }=1\right)$

The multiplicity of the polynomial $=-2\left(\text{Multiplicity }=3\right)$

To determine

To find:

b. To determine whether the graph crosses or touches the $x\text{-axis}$ at each $x\text{-intercept}$.

Answer to Problem 51AYU

b. $\text{Zero 1 has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

$\text{Zero }-1\text{ has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

$\text{Zero }-2\text{ has odd multiplicity. Therefore the graph crosses the }x\text{-axis}$.

Explanation of Solution

Given:

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3=4\left(x^2+1\right)\left(x^3-6x^2+12x-8\right)$

$=4\left(x^5-6x^4+13x^3-14x^2+12x-8\right)$

b. Zero $1$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

Zero $-1$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

Zero $-2$ has odd multiplicity. Therefore the graph crosses the $x\text{-axis}$.

To determine

To find:

c. To determine the maximum number of turning points on the graph.

Answer to Problem 51AYU

c. $\text{The maximum number of turning points }=\left(n-1\right)=4$.

Explanation of Solution

Given:

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3=4\left(x^2+1\right)\left(x^3-6x^2+12x-8\right)$

$=4\left(x^5-6x^4+13x^3-14x^2+12x-8\right)$

c. The degree of $f\left(x\right)=n=5$.

The maximum number of turning points $=\left(n-1\right)=4$.

To determine

To find:

d. To determine the end behavior (power function that the graph of $f$ resembles for large values of $x$).

Answer to Problem 51AYU

d. $\text{For large values of }x\text{ in the positive direction value of }f\text{ approaches positive infinity}$.

$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$

Explanation of Solution

Given:

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3$

$f\left(x\right)=4\left(x^2+1\right){\left(x-2\right)}^3=4\left(x^2+1\right)\left(x^3-6x^2+12x-8\right)$

$=4\left(x^5-6x^4+13x^3-14x^2+12x-8\right)$

d. $\text{f}\left(x\right)=4\left(x^5-6x^4+13x^3-14x^2+12x-8\right)$

The polynomial $f\left(x\right)=a_n{x}^n$.

Here $a_n=4>0$.

For large values of $x$ in the positive direction value of $f$ approaches positive infinity.

$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$