Problem 87 - Calculus Volume 1

Question

For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; \(c\) , find the \(y\) -intercept(s), if any; d. use the leading coefficient to determine the graph's end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. $$ f(x)=3 x-x^{3} $$

Step-by-Step Solution

Answer

a. Degree: 3; b. Zeros: 0, \(\pm \sqrt{3}\); c. Y-intercept: (0,0); d. End behavior: as \( x \to \infty, f(x) \to -\infty \); as \( x \to -\infty, f(x) \to \infty \); e. Function is odd.

Step 1: Find the Degree

The degree of a polynomial is the highest power of the variable in the polynomial. For the polynomial \( f(x) = 3x - x^3 \), the highest power of \( x \) is 3. Therefore, the degree of the polynomial is 3.

Step 2: Find the Zeros

To find the zeros of the polynomial, set the polynomial equal to zero and solve for \( x \). \[ 3x - x^3 = 0 \]Factor out the common factor \( x \):\[ x(3 - x^2) = 0 \]Set each factor equal to zero:1. \( x = 0 \)2. \( 3 - x^2 = 0 \, \Rightarrow \, x^2 = 3 \, \Rightarrow \, x = \pm \sqrt{3} \)Thus, the zeros are \( x = 0, \sqrt{3}, \text{and} -\sqrt{3} \).

Step 3: Find the Y-intercept

The \( y \)-intercept of a polynomial can be found by evaluating the polynomial at \( x = 0 \). For \( f(x) = 3x - x^3 \), substituting \( x = 0 \) gives:\[ f(0) = 3(0) - (0)^3 = 0 \]Thus, the \( y \)-intercept is \( (0, 0) \).

Step 4: Determine the End Behavior

The leading term of the polynomial \( f(x) = 3x - x^3 \) is \( -x^3 \), which has a negative coefficient and an odd degree. Therefore, as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

Step 5: Determine if the Polynomial is Even, Odd, or Neither

A function is even if \( f(-x) = f(x) \) for all \( x \), odd if \( f(-x) = -f(x) \), and neither if neither condition is met. Calculate \( f(-x) \):\[ f(-x) = 3(-x) - (-x)^3 = -3x + x^3 \]Compare with \( -f(x) \) which is:\[ -(3x - x^3) = -3x + x^3 \]Since \( f(-x) = -f(x) \), the function is odd.

Key Concepts

Polynomial Degree

The degree of a polynomial is the highest power of the variable within it. This is a crucial characteristic because it informs us about the behavior of the polynomial's graph.
In the polynomial function shown in the exercise, \( f(x) = 3x - x^3 \), the highest power of the variable \( x \) is 3.
This means that the degree of this polynomial is 3, which also tells us that the polynomial and its graph will have certain properties, like potential turning points and the maximum number of zeros.
Understanding the degree helps set expectations for how complex the polynomial's behavior will be as the input values vary.

Polynomial Zeros

Zeros of a polynomial, also known as roots or solutions, are the values of \( x \) that make the polynomial equal to zero. They are found by setting \( f(x) = 0 \) and solving for \( x \).
For our polynomial \( 3x - x^3 \), setting it equal to zero gives us:

• Factor out \( x \), resulting in \( x(3 - x^2) = 0 \).
• Solving \( x = 0 \), \( 3 - x^2 = 0 \) or \( x^2 = 3 \) reveals \( x = 0, \sqrt{3}, -\sqrt{3} \).

These zeros are the \( x \)-coordinates where the graphed function will intersect the \( x \)-axis. Knowing the zeros helps us understand basic graph positioning and intersections.

Y-Intercept

The \( y \)-intercept is where the graph of the polynomial crosses the \( y \)-axis, meaning it's where \( x = 0 \).
To find this intercept for \( f(x) \), we substitute \( x = 0 \):

• \( f(0) = 3(0) - (0)^3 = 0 \)

This results in a \( y \)-intercept of \( (0,0) \).
The \( y \)-intercept provides a starting point for sketching the graph and helps confirm the polynomial's behavior as it begins to take shape on the coordinate plane.

Leading Coefficient

The leading coefficient of a polynomial is part of the term with the highest degree and is essential for determining the end behavior of the polynomial's graph.
In \( f(x) = 3x - x^3 \), the leading term is \( -x^3 \), so the leading coefficient is \(-1\).
This negative leading coefficient, combined with the odd degree, means:

• As \( x \to \infty \), \( f(x) \to -\infty \).
• As \( x \to -\infty \), \( f(x) \to \infty \).

This behavior shows that one end of the graph will dip down while the other side rises up, based on the highest power term, setting overall shape expectations before detailed plotting.

Even and Odd Functions

Whether a polynomial is even, odd, or neither can influence its symmetry and graphical appearance.

• An even function satisfies \( f(-x) = f(x) \).
• An odd function satisfies \( f(-x) = -f(x) \).

For \( f(x) = 3x - x^3 \), we check:

• Calculate \( f(-x) = -3x + x^3 \).
• Compare this with \( -f(x) = -3x + x^3 \).

Since \( f(-x) = -f(x) \), this polynomial is odd.
This property suggests the graph is symmetric about the origin, giving more insight into its visual presentation.