Problem 20 - A Graphical Approach to College Algebra

Question

Suppose that a polynomial function \(P\) is defined in such a way that \(P(3)=-4\) and \(P(4)=-10 .\) Can we be certain that there is no zero between 3 and \(4 ?\) Explain, using a graph.

Step-by-Step Solution

Answer

No, we cannot be certain because both function values are negative.

Step 1: Understand the Problem

We need to determine if there is a zero for the polynomial function \( P(x) \) between \( x = 3 \) and \( x = 4 \). A zero means \( P(c) = 0 \) for some \( c \) between 3 and 4.

Step 2: Recall the Intermediate Value Theorem

The Intermediate Value Theorem states that for any continuous function \( f \) over a closed interval \([a, b]\), if \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in the interval \( (a, b) \) such that \( f(c) = 0 \).

Step 3: Analyze the Function Values

We know \( P(3) = -4 \) and \( P(4) = -10 \). Both values are negative, meaning they do not have opposite signs. Therefore, the Intermediate Value Theorem does not guarantee a zero in the interval \( [3, 4] \).

Step 4: Consider the Graph of \( P(x) \)

A polynomial function is continuous, but due to the lack of a sign change between \( P(3) \) and \( P(4) \), the graph does not necessarily cross the x-axis between these points. Hence, we can't be certain of a zero in \( [3, 4] \).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables and coefficients, subject to operations of addition, subtraction, multiplication, and non-negative integer exponents. Polyomials have different types, such as:

• Linear: These are polynomials of degree 1, like \( P(x) = 2x + 3 \).
• Quadratic: These are polynomials with degree 2, like \( Q(x) = x^2 + 5x + 6 \).
• Cubic and higher: Polynomials with degree 3 or more, like \( R(x) = x^3 - 2x^2 + 7 \).

The degree of a polynomial is the highest exponent of its variable. The function's graph gives us a good visual of where the function might cross the x-axis, but not all polynomials will have real roots. Understanding these characteristics is essential when predicting behavior over intervals.

Zero of a Function

The zero of a function, also known as the root, is a value of \( x \) that makes the function equal to zero. For example, if \( P(x) = 0 \), then the value \( x \) is a zero or a root of the function. Identifying zeros is crucial because:

• Graphical Representation: It's where the function crosses the x-axis.
• Solving Equations: Finding zeros can solve equations involving polynomials.
• Significance in Calculus: Roots can indicate intervals of interest for various calculus operations, like finding the area under curves.

In the given exercise, we were checking if the polynomial changes signs between two points (3 and 4). However, both \( P(3) \) and \( P(4) \) were negative, signaling no sign change, meaning there isn't necessarily a zero in between.

Continuous Function

A continuous function is one where small changes in the input produce small changes in the output. This means there are no gaps, jumps, or holes in the graph of the function. Polynomials are always continuous functions because:

• No Breaks or Gaps: Polynomials are defined for all real numbers without any interruptions.
• Smooth Graphs: Their graphs are smooth and unbroken lines or curves.
• Predictability: It supports the application of the Intermediate Value Theorem, which assists in understanding the behavior of function's zeros over specific intervals.

In our specific example, while the Intermediate Value Theorem helps with finding zeros by considering sign changes, continuous functions like polynomials do not guarantee zeros unless such sign changes are evident. Thus, a polynomial could be negative over an interval without crossing the x-axis.