Chapter 4.1, Problem 122AYU

Which of the following statements are true regarding the graph of the cubic polynomial $f\left(x\right)=x^3+b{x}^2+cx+d$? (Give reasons for your conclusions.)

a. It intersects the $y\text{-axis}$ in one and only one point.

b. It intersects the $x\text{-axis}$ in at most three points.

c. It intersects the $x\text{-axis}$ at least once.

d. For $\left|x\right|$ very large, it behaves like the graph of $y=x^3$.

e. It is symmetric with respect to the origin.

f. It passes through the origin.

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

a.

$\text{True}\text{. The }y\text{ intersect can be found by }f\left(x\right)=x^3+b{x}^2+cx+d$

$f\left(0\right)=d$

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

a.

It intersects the $y\text{-axis}$ in one and only one point

True. The $y\text{-intersect}$ can be found by $f\left(x\right)=x^3+b{x}^2+cx+d$

$f\left(0\right)=d$

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

b.

True.

$\text{The degree of the polynomial = 3}$.

$\text{Therefore there are 3 zeros of the polynomial and three}x\text{-intercept}$.

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

b.

It intersects the $x\text{-axis}$ at most three points.

True.

The degree of the polynomial $\text{= 3}$.

Therefore there are 3 zeros of the polynomial and three $x\text{-intercept}$.

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

c.

True.

$\text{The degree of the polynomial = 3}$.

$\text{Therefore there are 3 zeros of the polynomial and three}x\text{-intercept}$.

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

c.

It intersects the $x\text{-axis}$ at least once.

True.

The degree of the polynomial $\text{= 3}$.

Therefore there are 3 zeros of the polynomial and three $x\text{-intercept}$.

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

d.

$\text{For}\text{large values of }x\text{ the graph behaves like }{x}^3$.

True.

$\text{The end behavior of the cubic polynomial for large values of }x\text{is}{x}^3$.

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

d.

For large values of $x$ the graph behaves like $x^3$.

True.

The end behavior of the cubic polynomial for large values of $x$ is $x^3$.

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

e.

False

It is not symmetric with respect to the origin. Because the graph will have 2 turning points.

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

e.

False

It is not symmetric with respect to the origin. Because the graph will have 2 turning points.

To determine

To write: A cubic Polynomial function with certain characteristics.

Answer to Problem 122AYU

f.

It passes through the origin.

It might or might not pass through the origin.

Explanation of Solution

Given:

$f\left(x\right)=x^3+b{x}^2+cx+d$

f.

It passes through the origin.

It might or might not pass through the origin.