Problem 35 - Calculus

Question

Let \(p(x)\) be a cubic polynomial with \(p(5)<0, p(10)>\) \(0,\) and \(p(12)<0 .\) What can you say about the number and location of zeros of \(p(x) ?\)

Step-by-Step Solution

Answer

At least two zeros: one between 5 and 10, another between 10 and 12.

Step 1: Analyze Sign Changes

Given that \(p(5) < 0\), \(p(10) > 0\), and \(p(12) < 0\), we notice changes in the sign of the polynomial values. This change in sign indicates that there are zeros of the polynomial in the intervals where the sign changes.

Step 2: Identify Intervals with Zeros

From the conditions given, we see:- From \(x = 5\) to \(x = 10\), \(p(x)\) goes from negative to positive, indicating there is at least one zero in this interval.- From \(x = 10\) to \(x = 12\), \(p(x)\) goes from positive to negative, indicating at least another zero in this interval.

Step 3: Conclude on Location and Number of Zeros

Since \(p(x)\) is a cubic polynomial, it can have at most 3 real zeros. Based on the analysis:- There is definitely one zero between \(5\) and \(10\).- There is at least one more zero between \(10\) and \(12\).- There could potentially be a third zero in the same intervals or elsewhere before or after \(5\) or after \(12\), but the information provided doesn't specify.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial where the highest degree of the variable is three. In mathematical terms, it can be expressed as:

• \(p(x) = ax^3 + bx^2 + cx + d\)

Here, \(a, b, c,\) and \(d\) are constants, and \(a\) must be non-zero because if \(a\) were zero, the polynomial would not be cubic. Cubic polynomials are crucial as they can model real-world phenomena that exhibit some form of cubic growth or decay.
Cubic polynomials have various properties, one of which is the potential to have up to three real zeros. These zeros are the \(x\)-values where \(p(x) = 0\). A cubic polynomial is also continuous and differentiable, meaning it has no breaks or sharp corners in its graph, allowing for smooth transitions between different values.
The behavior of the function and its graph is largely determined by the leading coefficient \(a\). If \(a > 0\), the ends of the graph will trend in opposite directions: down to up as \(x\) moves from negative to positive infinity. Conversely, if \(a < 0\), the graph trends from up to down.

Sign Change

In the context of polynomial functions, a sign change refers to an alteration in the sign or direction of the polynomial's value as \(x\) changes. For example, if \(p(x)\) is negative at one point and positive at another, a sign change has occurred between those points. This is an indication that a zero must lie somewhere between those points.
Analyzing sign changes is an intuitive way to locate the intervals where zeros (roots) of the function might lie. Each interval where the polynomial changes sign suggests the presence of at least one zero, as the function must cross the \(x\)-axis from negative to positive or vice versa to produce that change.
When examining a cubic polynomial like in the exercise, we noticed sign changes between \(x = 5\) and \(x = 10\) and again between \(x = 10\) and \(x = 12\). This tells us that a root must exist in each of these intervals, validating our understanding of sign changes as signals of zeros.

Real Zeros

Real zeros of a polynomial are the values of \(x\) for which the polynomial equals zero, or when the graph of the polynomial intersects the \(x\)-axis. Since cubic polynomials can have at most three real zeros, these zeros are essential in determining the polynomial's roots.

• Real zeros are also critical to understand because they represent the solutions to the polynomial equation \(p(x) = 0\).

In the given exercise, by examining the sign changes, we deduced at least two real zeros—one between \(x = 5\) and \(x = 10\), and another between \(x = 10\) and \(x = 12\). Such analysis shows the crossing points on the graph between positive and negative values.
Zeros are not only theoretical solutions but also have practical applications in graphing, physics, engineering, and more. They help predict where certain events occur, such as when a moving object changes direction or when financial calculations hit a break-even point. In understanding cubic polynomials, identifying real zeros is crucial for a complete analysis of the function's behavior.