Problem 86 - 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. $$ f(x)=x^{3}+3 x^{2}-x-3 $$

Step-by-Step Solution

Answer

Degree 3, zeros -1, 1, -3, y-intercept -3, odd end behavior, neither.

Step 1: Identify the Degree of the Polynomial

To find the degree of the polynomial \(f(x) = x^3 + 3x^2 - x - 3\), we look for the term with the highest power of \(x\). Here, the highest power is 3, so the degree of the polynomial is 3.

Step 2: Find the Zeros of the Polynomial

To find the zeros of \(f(x) = x^3 + 3x^2 - x - 3\), we set the polynomial equal to zero: \(x^3 + 3x^2 - x - 3 = 0\). Checking simple rational roots through trial (using factors of constant term \(-3\)), we try \(x = -1, 1, 3, -3\). We find that \(x = -1\) satisfies \(f(x) = 0\) using the Rational Root Theorem. We then perform synthetic division by \(x + 1\), resulting in \(x^2 + 2x - 3\). Factoring \(x^2 + 2x - 3\) gives \((x-1)(x+3)\). Thus, the zeros are \(-1, 1,\) and \(-3\).

Step 3: Find the y-Intercept(s)

The \(y\)-intercept of a polynomial \(f(x)\) is the value of \(f(x)\) when \(x = 0\). Substitute \(x = 0\) into the polynomial: \(f(0) = (0)^3 + 3(0)^2 - 0 - 3 = -3\). Thus, the \(y\)-intercept is \( (0, -3) \).

Step 4: Determine the Graph's End Behavior

The end behavior of the polynomial is determined by the leading term, which is \(x^3\). Since the coefficient is positive and the degree is odd, as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity; as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.

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

A polynomial is even if \(f(-x) = f(x)\) for all \(x\), odd if \(f(-x) = -f(x)\), and neither if neither condition holds. Calculate \(f(-x)\): \[f(-x) = (-x)^3 + 3(-x)^2 - (-x) - 3 = -x^3 + 3x^2 + x - 3\]. \(-f(x)\) is \(-x^3 - 3x^2 + x + 3\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the polynomial is neither even nor odd.

Key Concepts

Degree of Polynomial

In mathematics, the degree of a polynomial is a crucial concept. It tells us the highest power of the variable present in the polynomial. In simpler terms, it's the biggest exponent of the variable among all the terms in the polynomial.

For instance, consider the polynomial \( f(x) = x^3 + 3x^2 - x - 3 \). Here, the term with the highest power of \( x \) is \( x^3 \). Therefore, the degree of this polynomial is 3. This indicates that it's a cubic polynomial.

Understanding the degree helps in predicting many features of the polynomial, like its graph and how many solutions or intersecting points it could potentially have.

Zeros of Polynomial

The zeros of a polynomial are the values of \( x \) for which the polynomial gives an output of zero. Think of them as the points where the graph of the polynomial crosses or touches the x-axis. Finding these zeros is critical in analyzing the behavior of the polynomial.

For \( f(x) = x^3 + 3x^2 - x - 3 \), we start by setting the polynomial equation to zero: \( x^3 + 3x^2 - x - 3 = 0 \). By testing possible values, often with the help of the Rational Root Theorem, we find \( x = -1 \) as a zero.

Using synthetic division with \( x + 1 \) shows that the quotient is \( x^2 + 2x - 3 \). Factoring this further reveals zeros at \( x = 1 \) and \( x = -3 \). Therefore, the zeros for this polynomial are \(-1, 1,\) and \(-3\).

End Behavior of Graphs

The end behavior of a polynomial’s graph describes how it behaves as \( x \) approaches positive or negative infinity. This is largely determined by the leading term of the polynomial, which is the term with the highest degree.

For \( f(x) = x^3 + 3x^2 - x - 3 \), the leading term is \( x^3 \). Because the coefficient of \( x^3 \) is positive and the degree is odd, as \( x \) approaches positive infinity, \( f(x) \) also heads towards positive infinity. Conversely, as \( x \) goes to negative infinity, \( f(x) \) tends towards negative infinity.

This behavior creates a distinct S-shaped curve typical of cubic polynomials, giving us insights into how the graph stretches and its directional trends.

Y-Intercept

The y-intercept of a polynomial is the point where its graph crosses the y-axis. To find it, substitute \( x = 0 \) into the polynomial and solve for \( f(x) \). This gives you the output when \( x \) is zero.

For our polynomial \( f(x) = x^3 + 3x^2 - x - 3 \), substituting \( x = 0 \) results in \( f(0) = (0)^3 + 3(0)^2 - 0 - 3 = -3 \). Thus, the y-intercept is at the point \( (0, -3) \).

The y-intercept is a key point on the graph, offering a starting clue about the graph’s position relative to the axes.

Even and Odd Functions

Determining whether a function is even, odd, or neither helps understand its symmetry. An even function satisfies \( f(-x) = f(x) \). This means it is symmetric with respect to the y-axis. An odd function meets the criteria \( f(-x) = -f(x) \), showing symmetry about the origin.

When examining \( f(x) = x^3 + 3x^2 - x - 3 \), compute \( f(-x) = (-x)^3 + 3(-x)^2 - (-x) - 3 = -x^3 + 3x^2 + x - 3 \). Compare this to \( -f(x) = -x^3 - 3x^2 + x + 3 \). Neither condition for evenness or oddness fits, meaning the function is neither even nor odd.

Understanding this aspect of a polynomial aids in predicting how its graph looks and behaves geometrically.