[Technology Exercise] Roots


Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.


c. It can be shown that the exact value of the solution in part (b) is


(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³


Evaluate this exact answer and compare it with the value you found in part (b).

Verified step by step guidance

1. First, let's understand the expression given: \( \left(\frac{1}{2} + \frac{\sqrt{69}}{18}\right)^{1/3} + \left(\frac{1}{2} - \frac{\sqrt{69}}{18}\right)^{1/3} \). This involves evaluating cube roots of two terms.
2. To evaluate this expression, start by calculating the values inside the cube roots separately. Compute \( \frac{1}{2} + \frac{\sqrt{69}}{18} \) and \( \frac{1}{2} - \frac{\sqrt{69}}{18} \).
3. Next, find the cube root of each of these values. This means calculating \( \left(\frac{1}{2} + \frac{\sqrt{69}}{18}\right)^{1/3} \) and \( \left(\frac{1}{2} - \frac{\sqrt{69}}{18}\right)^{1/3} \).
4. Once you have the cube roots, add them together to get the final result of the expression.
5. Finally, compare this result with the value you found in part (b) to see how they match or differ. This will help you understand the accuracy of your previous solution.

Key Concepts

Roots of a Polynomial

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. For the function ƒ(𝓍) = 𝓍³ - 𝓍 - 1, finding the roots involves solving the equation ƒ(𝓍) = 0. These roots can be real or complex and are essential for understanding the behavior of the polynomial function, including its intercepts and turning points.

Cubic Functions

Cubic functions are polynomial functions of degree three, characterized by their general form ƒ(𝓍) = 𝓍³ + ax² + bx + c. They can have one, two, or three real roots, depending on their discriminant. The shape of the graph of a cubic function can exhibit various behaviors, such as having inflection points and local maxima or minima, which are crucial for analyzing the function's properties.

Exact vs. Approximate Values

In calculus, the distinction between exact and approximate values is important for understanding solutions to equations. An exact value is a precise mathematical expression, such as (1/2 + √69/18)¹/³ + (1/2 - √69/18)¹/³, while an approximate value is a numerical estimate obtained through methods like numerical approximation or graphing. Comparing these values helps assess the accuracy of numerical methods used in solving equations.