Problem 10 - A Graphical Approach to College Algebra

Question

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{3}-x^{2}+5 x+5 ; \quad 2 \text { and } 3$$

Step-by-Step Solution

Answer

The function has a zero between 2 and 3, approximately at \( x \approx 2.28 \).

Step 1: Understanding the Given Function

We are given the polynomial function \( P(x) = -x^3 - x^2 + 5x + 5 \) and need to prove it has at least one real zero between \( x = 2 \) and \( x = 3 \). To apply the Intermediate Value Theorem, first evaluate the function at both points.

Step 2: Calculating \( P(2) \)

Substitute \( x = 2 \) into the function: \( P(2) = -(2)^3 - (2)^2 + 5(2) + 5 \). This simplifies to \( P(2) = -8 - 4 + 10 + 5 = 3 \). Thus, \( P(2) = 3 \).

Step 3: Calculating \( P(3) \)

Substitute \( x = 3 \) into the function: \( P(3) = -(3)^3 - (3)^2 + 5(3) + 5 \). This simplifies to \( P(3) = -27 - 9 + 15 + 5 = -16 \). Thus, \( P(3) = -16 \).

Step 4: Applying the Intermediate Value Theorem

The Intermediate Value Theorem states that if \( f(x) \) is continuous on \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \). Here, \( P(2) = 3 \) and \( P(3) = -16 \), which have opposite signs. Thus, there is at least one real zero in the interval \((2, 3)\).

Step 5: Approximating the Zero Using a Calculator

Using a calculator, find the point where \( P(x) = 0 \) between \( x = 2 \) and \( x = 3 \). Enter the function into the graphing calculator and use numerical methods like the "zero" function to determine the zero. The approximate zero is \( x \approx 2.28 \).

Key Concepts

Real Zero

In the context of a polynomial function, a real zero refers to a value of \( x \) where the function equals zero. This is essentially the point where the graph of the polynomial crosses or touches the \( x \)-axis. Real zeros are crucial for understanding the behavior of polynomial functions, as they help to identify critical points where the value of the function changes sign. When dealing with continuous functions like polynomials, finding a real zero often involves evaluating the function at different points to see where the signs of the function values change. This change of sign is instrumental because it indicates a transition from positive to negative values, or vice versa, suggesting the existence of a real zero. In our example, by applying the Intermediate Value Theorem, we found that there is a change in the sign of the function from \( P(2) = 3 \) to \( P(3) = -16 \). This sign change between \( x = 2 \) and \( x = 3 \) confirmed the presence of at least one real zero in this interval.

Polynomial Function

A polynomial function is a mathematical expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial is:\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]Where:

• \( a_n, a_{n-1}, ..., a_1, a_0 \) are constants called coefficients
• \( n \) is a non-negative integer

In the given exercise, we analyzed the polynomial function \( P(x) = -x^3 - x^2 + 5x + 5 \). Each term in the function is a product of a constant coefficient and a power of \( x \). Polynomial functions are continuous, which means they have no breaks, holes, or jumps in their graphs. This property makes them especially suitable for application of theorems like the Intermediate Value Theorem, as used in the exercise to find real zeros. Understanding how polynomial functions behave allows us to explore their roots, direction (whether they are increasing or decreasing), and other properties significant in calculus.

Numerical Approximation

Numerical approximation is a mathematical technique used to find approximate solutions to real problems. When exact answers are difficult or impossible to determine using algebraic methods, numerical techniques provide a practical solution by estimating values to a specified level of accuracy.In the case of finding a real zero for the given polynomial function, numerical approximation can be useful after establishing that a zero exists between two values. For our exercise, after applying the Intermediate Value Theorem, we used a numerical method, likely via a calculator, to approximate the zero. These methods often include:

• Graphing techniques, where you visually inspect where the function crosses the \( x \)-axis
• Navigating functions on calculators, such as the "zero" function or root-finding tools

For the polynomial function \( P(x) \), an approximation approach yielded an approximate real zero at \( x \approx 2.28 \), showing the practical application of such methods. Numerical approximation helps in precisely pinpointing where the transition from positive to negative (or vice versa) occurs, giving us the estimated location of the zero with a degree of confidence.