Problem 13 - Precalculus

Question

Determine whether the given value is a zero of the function. \(f(x)=2 x^{3}-3 x+1\) (a) \(x=(\sqrt{3}-1) / 2\) (b) \(x=(\sqrt{3}+1) / 2\)

Step-by-Step Solution

Answer

Neither value is a zero of the function.

Step 1: Understand the Problem

To determine if a given value is a zero of a function, we need to substitute the value into the function and verify if the output is zero. If the result is zero, then the given value is a zero of the function.

Step 2: Substitution for (a)

Substitute \(x = \frac{\sqrt{3} - 1}{2}\) into the function \(f(x) = 2x^3 - 3x + 1\). Calculate \[f\left(\frac{\sqrt{3} - 1}{2}\right) = 2\left(\frac{\sqrt{3} - 1}{2}\right)^3 - 3\left(\frac{\sqrt{3} - 1}{2}\right) + 1\].

Step 3: Evaluate the Substitution for (a)

Calculate each term: - First term: \(2\left(\frac{\sqrt{3} - 1}{2}\right)^3 = 2\left(\frac{(\sqrt{3} - 1)^3}{8}\right)\)- Expand \((\sqrt{3} - 1)^3 = \sqrt{3}^3 - 3 \cdot \sqrt{3}^2 \cdot 1 + 3 \cdot \sqrt{3} \cdot 1^2 - 1^3 = 3\sqrt{3} - 9 + 3\sqrt{3} - 1\)- Sum up: \(2\left(\frac{6\sqrt{3} - 10}{8}\right)\)- Second term: \(- 3\left(\frac{\sqrt{3} - 1}{2}\right)\)- Simplify and calculate both terms.

Step 4: Substitution for (b)

Substitute \(x = \frac{\sqrt{3} + 1}{2}\) into the function \(f(x) = 2x^3 - 3x + 1\). Calculate \[f\left(\frac{\sqrt{3} + 1}{2}\right) = 2\left(\frac{\sqrt{3} + 1}{2}\right)^3 - 3\left(\frac{\sqrt{3} + 1}{2}\right) + 1\].

Step 5: Evaluate the Substitution for (b)

Calculate each term: - First term: \(2\left(\frac{\sqrt{3} + 1}{2}\right)^3 = 2\left(\frac{(\sqrt{3} + 1)^3}{8}\right)\)- Expand \((\sqrt{3} + 1)^3 = \sqrt{3}^3 + 3 \cdot \sqrt{3}^2 \cdot 1 + 3 \cdot \sqrt{3} \cdot 1^2 + 1^3 = 3\sqrt{3} + 9 + 3\sqrt{3} + 1\)- Sum up: \(2\left(\frac{6\sqrt{3} + 10}{8}\right)\)- Second term: \(- 3\left(\frac{\sqrt{3} + 1}{2}\right)\)- Simplify and calculate both terms.

Step 6: Combine Results for Both (a) and (b)

By evaluating the calculations, determine if \(f\left(\frac{\sqrt{3} - 1}{2}\right)\) or \(f\left(\frac{\sqrt{3} + 1}{2}\right)\) equals zero. This will reveal if either of the given values is a zero of the function.

Key Concepts

Polynomial Function

Polynomial functions are a central topic in algebra and calculus, representing expressions consisting of variables and coefficients. These functions come in various degrees and forms, such as linear, quadratic, and cubic polynomials, each determined by the highest power of the variable involved.

Key characteristics of polynomial functions include:

• A variable raised to whole-number exponents, which creates terms like \(x^3\), \(x^2\), etc.
• Real number coefficients such as 2, -3, or 1, as seen in the function \(f(x) = 2x^3 - 3x + 1\).
• A behavior that can be studied to understand the function's roots (zeros), which are the points where it crosses or touches the x-axis on a graph.

Identifying the degree of the polynomial is crucial since it hints at the number of potential real zeros. A cubic function, like \(2x^3 - 3x + 1\), suggests it might have up to three real zeros, guiding our problem-solving strategy.

Substitution Method

The substitution method is a straightforward technique to solve polynomial equations, particularly useful in testing whether a specific value is a zero of a function.
The process involves:

• Replacing, or substituting, the variable \(x\) in the polynomial with the given value. For instance, when testing \(x = \frac{\sqrt{3} - 1}{2}\) in the function \(f(x) = 2x^3 - 3x + 1\), you substitute \(x\) with \(\frac{\sqrt{3} - 1}{2}\).
• Calculating the result, which determines whether or not it is zero. If the outcome is zero, the substituted value is a zero of the function.
• This method, as shown, requires simplifying several terms and can involve arithmetic with fractions or radicals, demanding careful calculation to reach an accurate conclusion.

This method is particularly insightful as it confirms or refutes the zero status of any given value, directly linking algebraic manipulation to function behavior.

Cubic Function Zero Test

The cubic function zero test is applied to determine whether specific numbers are zeros of a cubic polynomial. A cubic function, such as \(f(x) = 2x^3 - 3x + 1\), may offer three potential zeros since a degree of three implies there could be up to three values of \(x\) where the function equals zero.
Steps involved in the zero test include:

• Using the substitution method to check each potential zero individually. This reveals what you have found and calculated appropriately, leading to zero or another value.
• Considering the nature of cubic equations that might lead to multiple real and complex roots, although our focus typically remains on real values given in exercises.

Successfully identifying zeros helps in graphing the polynomial, understand its symmetry and behavior over its domain, and provides strategic insights into more complex computations like factoring the polynomial or solving equations.This approach highlights the interplay of algebraic techniques and function analysis in finding solutions.