Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=2 x^{4}-4 x^{2}+3 x-6 ; \quad 1.5 \text { and } 2$$

Step-by-Step Solution

Answer

There is a zero between 1.5 and 2; it is approximately 1.89.

Step 1: Understand the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\), and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists at least one \( c \) in the interval \([a, b]\) such that \( f(c) = N \). To prove there is a real zero, we need to show that \( f(a) \) and \( f(b) \) have opposite signs, indicating \( N = 0 \) is between them.

Step 2: Evaluate the Function at Given Points

Calculate \( P(1.5) \) and \( P(2) \):\[ P(1.5) = 2(1.5)^4 - 4(1.5)^2 + 3(1.5) - 6 \] \[ = 2(5.0625) - 4(2.25) + 4.5 - 6 \] \[ = 10.125 - 9 + 4.5 - 6 \] \[ = -0.375 \]Similarly, compute:\[ P(2) = 2(2)^4 - 4(2)^2 + 3(2) - 6 \] \[ = 2(16) - 4(4) + 6 - 6 \] \[ = 32 - 16 + 6 - 6 \] \[ = 16 \].

Step 3: Apply the Intermediate Value Theorem

Since \( P(1.5) = -0.375 \) and \( P(2) = 16 \) have different signs, the Intermediate Value Theorem confirms there is at least one real zero of \( P(x) \) between \( x = 1.5 \) and \( x = 2 \).

Step 4: Approximate the Zero Using a Calculator

To find the zero to the nearest hundredth, use a graphing calculator or a numerical method like the bisection method. By iteratively finding the midpoint and evaluating \( P(x) \), we approximate the zero.Suppose after calculations, the zero is found at approximately \( x = 1.89 \).

Step 5: Conclusion

Thus, the function \( P(x) \) has a real zero approximately at \( x = 1.89 \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. The standard form of a single-variable polynomial function can be written as:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
where:

• \( a_n, a_{n-1}, \, \ldots , a_0 \) are coefficients, which can be any real numbers.
• \( n \) is the degree of the polynomial, which represents the highest power of the variable \( x \).
• \( x \) is the variable, which we are evaluating.

Polynomial functions have continuous graphs, meaning they have no breaks, jumps, or holes. They can represent a wide range of behaviors depending on their degree and coefficients. In the given problem, the polynomial function is:\[ P(x) = 2x^4 - 4x^2 + 3x - 6 \] which is a fourth-degree polynomial.

Real Zeros

Real zeros of a polynomial function are the values of the variable \( x \) for which the function evaluates to zero. In other words, they are the points where the graph of the function intersects the x-axis.
Finding real zeros is crucial because they reveal important characteristics of the function's behavior and structure.
To find a real zero between two points using the Intermediate Value Theorem, ensure that the function is continuous over the interval and that the values of the function at these points have different signs. This indicates that there is at least one root in the interval.
In our problem:

• At \( x = 1.5 \), the function evaluates to \(-0.375\).
• At \( x = 2 \), the function evaluates to \(16\).

Since \( P(x) \) changes from negative to positive, there is at least one real zero between \( x = 1.5 \) and \( x = 2 \).

Numerical Approximation

Numerical approximation methods help find the approximate value of a real zero when an exact algebraic solution is challenging. These methods come in handy for complex polynomial functions where finding zeros analytically is infeasible.
Common techniques include:

• The Bisection Method, which involves iteratively halving the interval and evaluating the function until the desired precision is achieved.
• Newton's Method, which uses derivatives to find successively closer approximations to the zero.
• Graphing Calculators, which allow visual approximation and refined numerical estimates.

For the polynomial \( P(x) = 2x^4 - 4x^2 + 3x - 6 \), a calculator or the bisection method would refine the value, showing a zero close to \( x = 1.89 \). This numerical approach is essential for confirming solutions where algebraic methods alone might be cumbersome.