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Finding the roots of a polynomial is essential to understand the points where it crosses the x-axis. For the Legendre polynomial given by:\[ P(x) = \frac{1}{2} \left(5x^3 - 3x\right) \]Finding the roots involves setting the polynomial equal to zero:\[ \frac{1}{2}x(5x^2 - 3) = 0 \]To solve this, you break it down into simpler equations:

• \( x = 0 \)
• \( 5x^2 - 3 = 0 \)

The equation \(5x^2 - 3 = 0\) is solved by moving 3 to the other side and dividing by 5, giving:\[ x^2 = \frac{3}{5} \]Now, take the square roots, noting both positive and negative solutions:\[ x = \pm\sqrt{\frac{3}{5}} \]So, the roots of our polynomial are \( x = 0, \sqrt{\frac{3}{5}}, \text{and} -\sqrt{\frac{3}{5}} \). These points help us understand where the polynomial crosses the x-axis, which is vital for graphing.

Graphing polynomials becomes more intuitive once you know the roots. These roots divide the graph into regions or intervals. For our Legendre polynomial:\( P(x) = \frac{1}{2}x(5x^2 - 3) \), we have roots at \( x = 0 \), \( x = \sqrt{\frac{3}{5}}\), and \( x = -\sqrt{\frac{3}{5}} \). The sign of the polynomial in each interval is determined by selecting test points within them:

• When \( x < -\sqrt{\frac{3}{5}} \), \( P(x) < 0 \).
• When \(-\sqrt{\frac{3}{5}} < x < 0 \), \( P(x) > 0 \).
• When \( 0 < x < \sqrt{\frac{3}{5}} \), \( P(x) < 0 \).
• When \( x > \sqrt{\frac{3}{5}} \), \( P(x) > 0 \).

Each interval shows a different behavior for the polynomial. The curve crosses the x-axis at each root. The graph's general shape is cubic, reflecting the degree of the polynomial and showing an inflection point near \( x = 0 \). When sketching the polynomial, start from the leftmost interval and follow the determined behavior to ensure accurate representation.

Interval notation is a concise way of expressing the set of all possible values of \( x \) for which a polynomial either stays positive or negative. We use these intervals to specify where the polynomial \( P(x) = \frac{1}{2}(5x^3 - 3x) \) is greater than or less than zero.The roots you've calculated split the x-axis into specific intervals:

• \((-\infty, -\sqrt{\frac{3}{5}})\) : Here, \( P(x) < 0 \)
• \((-\sqrt{\frac{3}{5}}, 0)\) : Here, \( P(x) > 0 \)
• \((0, \sqrt{\frac{3}{5}})\) : Here, \( P(x) < 0 \)
• \((\sqrt{\frac{3}{5}}, \infty)\) : Here, \( P(x) > 0 \)

These intervals show where the function increases or decreases and are crucial for solving inequalities and understanding the graph's behavior. Always use a parenthesis when the endpoint isn’t included, which is typical when dealing with open intervals in polynomial inequalities.