Problem 39 - Precalculus: Functions and Graphs

Question

A lesendre polymomial The third-degree Legendre polynomial \(P(x)=\frac{1}{2}\left(5 x^{3}-3 x\right)\) occurs in the solution of heat transfer problems in physics and engineering. Find all values of \(x\) such that \(P(x)>0\) and all \(x\) such that \(P(x)<0,\) and sketch the graph of \(P\).

Step-by-Step Solution

Answer

\(P(x)>0\) for \((-\frac{\sqrt{15}}{5}, 0)\) and \((0, \frac{\sqrt{15}}{5})\); \(P(x)<0\) for \((-\infty, -\frac{\sqrt{15}}{5})\) and \((\frac{\sqrt{15}}{5}, \infty)\).

Step 1: Understand the Polynomial Function

The polynomial given is \( P(x) = \frac{1}{2} (5x^3 - 3x) \). This is a cubic polynomial, which means it will have up to three roots and can change signs at these roots.

Step 2: Find the Roots

To find the roots, solve the equation \( P(x) = 0 \). Set \( \frac{1}{2} (5x^3 - 3x) = 0 \). Simplify to \( 5x^3 - 3x = 0 \). Factor the equation: \( x(5x^2 - 3) = 0 \). The solutions are \( x = 0 \) or solve \( 5x^2 - 3 = 0 \) for other values.

Step 3: Solve the Quadratic Equation

Solve \( 5x^2 - 3 = 0 \). This gives \( 5x^2 = 3 \), or \( x^2 = \frac{3}{5} \). Thus, \( x = \pm \sqrt{\frac{3}{5}} = \pm \frac{\sqrt{15}}{5} \). Hence, the roots are \( x = 0, \pm \frac{\sqrt{15}}{5} \).

Step 4: Determine the Intervals for \(P(x) > 0\) and \(P(x) < 0\)

The roots split the real line into intervals. Check the sign of \( P(x) \) in each interval: \((-\infty, -\frac{\sqrt{15}}{5})\), \((-\frac{\sqrt{15}}{5}, 0)\), \((0, \frac{\sqrt{15}}{5})\), and \((\frac{\sqrt{15}}{5}, \infty)\). Choose a test point in each to determine the sign of \( P(x) \).

Step 5: Test Values in Each Interval

Choose test points: for \((-\infty, -\frac{\sqrt{15}}{5})\), use \(-1\); for \((-\frac{\sqrt{15}}{5}, 0)\), use \(-0.1\); for \((0, \frac{\sqrt{15}}{5})\), use \(0.1\); for \((\frac{\sqrt{15}}{5}, \infty)\), use \(1\). Substitute these values into \( P(x) = \frac{1}{2} (5x^3 - 3x) \) to determine sign.

Step 6: Interpret the Results

From the test points: \( P(-1) < 0 \), \( P(-0.1) > 0 \), \( P(0.1) > 0 \), and \( P(1) < 0 \). Thus, \( P(x) > 0 \) on the intervals \((-\frac{\sqrt{15}}{5}, 0)\) and \((0, \frac{\sqrt{15}}{5})\). \( P(x) < 0 \) on intervals \((-\infty, -\frac{\sqrt{15}}{5})\) and \((\frac{\sqrt{15}}{5}, \infty)\).

Step 7: Sketch the Graph of \(P(x)\)

Plotting the function considering the roots and the sign changes, the graph crosses the x-axis at \( x = 0 \) and \( \pm \frac{\sqrt{15}}{5} \), creating a cubic shape with maxima and minima between the roots.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial that has a degree of three, meaning the highest power of the variable, often represented by \(x\), is three. In the case of the Legendre polynomial given above, \(P(x) = \frac{1}{2} (5x^3 - 3x)\), the leading term is \(5x^3\). Cubic polynomials can have up to three roots and may exhibit up to two turning points, which could be a maximum and minimum point.
To identify important characteristics of cubic polynomials, you should note:

• Shape: Cubic functions generally have an "S" shape or an upside-down "S" shape depending on the sign of the leading coefficient. Positive leading coefficients suggest an upward opening, while negative suggest downward.
• Number of Roots: A cubic polynomial can have three real roots, one real and two complex conjugate roots, or just one real root repeated three times depending on its discriminant.

Understanding these properties will help in analyzing and graphing the polynomial sensibly.

Roots of Polynomial

Finding the roots of a polynomial is the process of solving the equation where the polynomial equals zero. Here, the roots are the solutions to \(P(x) = \frac{1}{2} (5x^3 - 3x) = 0\). Let's break down the steps:

• Factoring: Start by taking out the common factor, \(x\), which gives us \(x(5x^2 - 3) = 0\). This reveals one root at \(x = 0\).
• Solving the Quadratic: Next, set the quadratic expression \(5x^2 - 3 = 0\). Solving this yields roots at \(x = \pm \frac{\sqrt{15}}{5}\).

These roots are crucial because they determine where the polynomial will intersect the x-axis and split the graph into regions where the function's behavior changes. Thus, they are pivotal in understanding the changes in sign that inform about the positivity or negativity of the function over different intervals.

Graph Sketching

Graph sketching involves plotting a rough visualization of the polynomial function based on calculated roots and the intervals found from testing the function's sign. When graphing \(P(x)\) from \(P(x) = \frac{1}{2}(5x^3 - 3x)\), follow these guidelines:

• Plot the Roots: As established, plot points at \(x = 0\), \(x = \pm \frac{\sqrt{15}}{5}\). These marks where the graph intersects or touches the x-axis.
• Identify Turns: Between these points, the cubic nature implies one curve from negative to positive and back. Usually, these turning points involve local maxima and minima.
• Check Behavior at Infinities: Determine the directions as \(x\) approaches \(\pm\infty\), which in this polynomial, aligns with the cubic's "S" shape.

These steps ensure that you form a reasonable sketch, representing the key features and behaviors of the polynomial. Combined with calculated sign changes, it lays the groundwork for further calculations or interpretations.

Sign Change of Function

The sign change of a function relates to how the function moves from positive to negative values or vice versa. It's key in determining the intervals over which the function, like our Legendre polynomial, is positive or negative.

• Identify Intervals: Use the roots \(x = 0\) and \(x = \pm \frac{\sqrt{15}}{5}\) to split the number line into segments. Here, these are: \(( -\infty, -\frac{\sqrt{15}}{5})\), \(( -\frac{\sqrt{15}}{5}, 0)\), \((0, \frac{\sqrt{15}}{5})\), and \((\frac{\sqrt{15}}{5}, \infty)\).
• Test Points: Select test points in each interval (e.g., \(-1, -0.1, 0.1, 1\)) and substitute into the polynomial to check the sign of \(P(x)\).
• Conclusion on Signs: From tests: \(P(x) > 0\) in \((-\frac{\sqrt{15}}{5}, 0)\) and \((0, \frac{\sqrt{15}}{5})\), and \(P(x) < 0\) in \(( -\infty, -\frac{\sqrt{15}}{5})\) and \(( \frac{\sqrt{15}}{5}, \infty)\).

Understanding where a function changes sign prepares you to tackle problems involving constraints or other mathematical operations grounded in domain-specific conditions.