Problem 87 - Calculus Volume 1

Question

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

Degree: 3. Zeros: 0, \(\sqrt{3}\), \( -\sqrt{3}\). End behavior: \(-x^3\) causes graph to fall left, rise right. Polynomial is odd.

Step 1: Determine the Degree of the Polynomial

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

Step 2: Find the Zeros of the Polynomial

To find the zeros of the polynomial, set \(f(x) = 0\):\[3x - x^3 = 0\]Factor out \(x\):\[x(3 - x^2) = 0\]This gives the equations \(x = 0\) or \(3 - x^2 = 0\). Solving \(3 - x^2 = 0\) gives \(x^2 = 3\), thus \(x = \sqrt{3}\) or \(x = -\sqrt{3}\). So, the zeros are \(x = 0, \sqrt{3}, -\sqrt{3}\).

Step 3: Find the y-intercept

The \(y\)-intercept of a polynomial is found by evaluating \(f(0)\). Substituting \(x = 0\) into \(f(x)\), we get:\[f(0) = 3(0) - (0)^3 = 0\]Thus, the \(y\)-intercept is \(0\).

Step 4: Determine the End Behavior

The end behavior of a polynomial is determined by the leading term, which is the term with the highest degree. Since the leading term is \(-x^3\) and the leading coefficient is negative, as \(x\) approaches \(\infty\), \(f(x)\) goes to \(-\infty\), and as \(x\) approaches \(-\infty\), \(f(x)\) approaches \(\infty\).

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

To determine whether the polynomial is even or odd, use the definitions: a polynomial is even if \(f(-x) = f(x)\) for all \(x\), and odd if \(f(-x) = -f(x)\). For \(f(x) = 3x - x^3\), calculate \(f(-x)\):\[f(-x) = 3(-x) - (-x)^3 = -3x + x^3 = -(3x - x^3) = -f(x)\]Since \(f(-x) = -f(x)\), the polynomial is odd.

Key Concepts

Polynomial Degree

The degree of a polynomial provides critical information about its nature. It is defined as the highest power of the variable present in the polynomial expression. In our example with the polynomial \(f(x) = 3x - x^3\), we identify the term \(-x^3\) as having the highest power, which is 3. So, the degree of this polynomial is 3. Understanding the degree helps us determine several properties about the polynomial. For instance, the degree tells us the maximum number of zeros the polynomial might have. It also suggests the number of turning points the graph of the polynomial can have, which is usually one less than the degree.

Finding Zeros

Finding the zeros of a polynomial involves setting the polynomial equal to zero and solving for the variable. These values are the points where the graph of the polynomial crosses or touches the x-axis. For our polynomial \(f(x) = 3x - x^3\), we set it equal to zero:\[3x - x^3 = 0\]To solve, we factor out the common term, \(x\):\[x(3 - x^2) = 0\]This leads to two simple equations: \(x = 0\) and \(3 - x^2 = 0\). Solving the second equation gives us \(x^2 = 3\), resulting in \(x = \pm\sqrt{3}\). Therefore, the zeros are \(x = 0, \sqrt{3}, -\sqrt{3}\). These are the values where the polynomial intersects the x-axis.

End Behavior

The end behavior of a polynomial describes how the polynomial behaves as the variable approaches very large positive or negative values. It's primarily determined by the leading term, which in this case is \(-x^3\). A few key points to remember about end behavior:

• If the leading coefficient is positive, and the degree is odd, the polynomial falls to the left and rises to the right.
• If the leading coefficient is negative, and the degree is odd, the polynomial rises to the left and falls to the right.

For our polynomial \(-x^3\), the leading coefficient is negative, and the degree is odd (3), thus:- As \(x\) approaches \(\infty\), \(f(x)\) approaches \(-\infty\).- As \(x\) approaches \(-\infty\), \(f(x)\) approaches \(\infty\).

Even and Odd Functions

Determining if a polynomial is even, odd, or neither requires checking how the function responds to negative inputs. An even polynomial satisfies \(f(-x) = f(x)\), meaning it is symmetric across the y-axis. An odd polynomial satisfies \(f(-x) = -f(x)\), meaning it is symmetric with respect to the origin. To find out whether \(f(x) = 3x - x^3\) is even or odd, first calculate \(f(-x)\):\[f(-x) = 3(-x) - (-x)^3 = -3x + x^3\]This simplifies to:\[-(3x - x^3) = -f(x)\]Since \(f(-x) = -f(x)\), \(f(x)\) is confirmed to be an odd function. This indicates that the graph of \(f(x)\) has rotational symmetry about the origin.