Problem 37 - College Algebra

Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{3}+x^{2}-2 x+1 ; \text { between }-3 \text { and }-2 $$

Step-by-Step Solution

Answer

The function \(f(x)\) has at least one real zero between -3 and -2, as demonstrated by evaluation at the bounds and the application of the Intermediate Value Theorem.

Step 1: Function evaluation at interval boundaries

Evaluate the function \(f(x)\) at \(x=-3\) and \(x=-2\). If \(f(-3) = k1\) and \(f(-2) = k2\), we must find \(k1\) and \(k2\).

Step 2: Check the signs of function values

Check the values of \(k1\) and \(k2\). If \(k1\) and \(k2\) have opposite signs (meaning one is positive and the other is negative), then there must be at least one root of the function in the interval between -3 and -2. If not, then the Intermediate Value Theorem cannot guarantee a root within this range.

Step 3: Application of the Intermediate Value Theorem

If \(k1\) and \(k2\) bear opposite signs, it means that at some point between -3 and -2, the function \(f(x)\) must cross the x-axis (a zero exists). This is because the function \(f(x)\) is continuous (because it is a polynomial), and a continuous function cannot 'jump' from a positive value to a negative value (or vice versa) without crossing the x-axis. By the Intermediate Value Theorem, we can then conclude that there exists a real zero of \(f(x)\) between -3 and -2.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined with coefficients. These functions can be written in the general form:
\[\begin{equation} f(x) = a_n x^n + a_{n-1} x^{n-1} + \.\.\. + a_2 x^2 + a_1 x + a_0\text{
}where \text{
}n \text{ is a nonnegative integer and each } a_i \text{ is a constant coefficient}.\text{
}\end{equation}\]In our example, the polynomial function is
\[\begin{equation} f(x) = x^3 + x^2 - 2x + 1\text{
}\end{equation}\]which is a cubic polynomial because the highest degree of the variable is three.

Characteristics of Polynomial Functions

• Continuous: They have no breaks, jumps, or holes in their graphs.
• Differentiable: They have slopes (derivatives) that can be calculated at all points in their domain.
• End behavior: Depending on the degree and the leading coefficient, the ends of the graph will either rise or fall as the variable approaches infinity or negative infinity.

Polynomials are fundamental in mathematics because they can approximate many other functions and have nice properties that make them manageable to work with analytically. The function given in the exercise demonstrates these attributes and serves as a typical example of polynomial behavior.

Real Zeros of Polynomials

The real zeros of a polynomial function are the x-values for which the function yields a zero value. In other words, they are the roots or solutions of the equation where the polynomial is set equal to zero:
\[\begin{equation} f(x) = 0\text{
}\end{equation}\]Finding these zeros is crucial in understanding the behavior of polynomials and in solving polynomial equations. A polynomial of degree n has n complex roots, according to the Fundamental Theorem of Algebra, and some of these may be real zeros.

Locating Real Zeros

• Graphical Method: Plotting the polynomial function and identifying where it crosses the x-axis.
• Analytic Method: Using various algebraic techniques, like factorization, to find the zeros.
• Numerical Method: Applying algorithms to approximate the zeros to a desired precision.

In the context of the Intermediate Value Theorem, if we have a continuous polynomial function that changes sign over an interval, then there must be at least one real zero within that interval. The step-by-step solution shows how to apply this theorem to conclude the presence of a real zero between two integers.

Function Continuity

Function continuity refers to a function that is unbroken or uninterrupted across its domain. For a function to be continuous at a specific point, three conditions must be met:

• The function is defined at that point.
• The limit of the function as it approaches the point from both sides exists.
• The function's value at that point is equal to the limit.

For polynomial functions like the one in our exercise, continuity is a given across the entire set of real numbers. This is why we can use powerful tools like the Intermediate Value Theorem to analyze their behavior.

Why Continuity Matters

Continuity ensures that the function behaves predictably and smoothly, making it possible to use calculus techniques, such as finding limits, derivatives, and integrals. When applied in conjunction with the Intermediate Value Theorem, as seen in our example, it guarantees that if a continuous function changes sign over an interval, the function must pass through zero. This is a foundational concept that underpins many principles in calculus and aids significantly in the analysis of functions.