Chapter 4.1, Problem 124AYU

The illustration shows the graph of a polynomial function.

a. Is the degree of the polynomial even or odd?

b. Is the leading coefficient positive or negative?

c. Is the function even, odd, or neither?

d. Why is $x^2$ necessarily a factor of the polynomial?

e. What is the minimum degree of the polynomial?

f. Formulate five different polynomials whose graphs could look like the one shown. Compare yours to those of other students. What similarities do you see? What differences?

To determine

To find: The illustration shows the graph of a polynomial function:

a. Is the degree of the polynomial even or odd?

Answer to Problem 124AYU

a. The degree of the polynomial is even since it is symmetric with respect to the $y\text{-axis}$.

Explanation of Solution

Given:

a. The degree of the polynomial is even since it is symmetric with respect to the $y\text{-axis}$.

To determine

To find: The illustration shows the graph of a polynomial function:

b. Is the leading coefficient positive or negative?

Answer to Problem 124AYU

b. The leading coefficient is positive.

Explanation of Solution

Given:

b. The leading coefficient is positive.

To determine

To find: The illustration shows the graph of a polynomial function:

c. Is the function even, odd, or neither?

Answer to Problem 124AYU

c. The function is odd.

Explanation of Solution

Given:

c. The function is odd.

To determine

To find: The illustration shows the graph of a polynomial function:

d. Why is $x^2$ necessarily a factor of the polynomial?

Answer to Problem 124AYU

d. $x^2$ is necessary because the graph opens up on both the sides.

Explanation of Solution

Given:

d. $x^2$ is necessary because the graph opens up on both the sides.

To determine

To find: The illustration shows the graph of a polynomial function:

e. What is the minimum degree of the polynomial?

Answer to Problem 124AYU

e. The minimum degree of the polynomial is 4.

Explanation of Solution

Given:

e. The minimum degree of the polynomial is 4.

To determine

To find: The illustration shows the graph of a polynomial function:

f. Formulate five different polynomials whose graphs could look like the one shown. Compare yours to those of other students. What similarities do you see? What differences?

Answer to Problem 124AYU

f. $f\left(x\right)=x^4+b{x}^3+c{x}^2+dx+e, f\left(x\right)=x^4-b{x}^3-c{x}^2-dx+e, f\left(x\right)=x^4+b{x}^3-c{x}^2-dx+e$

Explanation of Solution

Given:

f. $f\left(x\right)=x^4+b{x}^3+c{x}^2+dx+e, f\left(x\right)=x^4-b{x}^3-c{x}^2-dx+e, f\left(x\right)=x^4+b{x}^3-c{x}^2-dx+e$