Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

By plugging in 1 and 2 into the function \(f(x) = x^{3}-x-1\), we get \(f(1) = -1\) and \(f(2) = 3\). It's clear that \(f(1)\) and \(f(2)\) have different signs. Therefore, by the Intermediate Value Theorem, because the function is continuous and the function values at the endpoints of the interval [1, 2] are of opposite signs, there is at least one real zero between 1 and 2 for the given polynomial.

Step 1: Evaluate the function at the given points

Evaluate the function \( f(x) = x^{3}-x-1 \) at \( x = 1 \) and \( x = 2 \). This gives you two values, \( f(1) \) and \( f(2) \).

Step 2: Check the signs of \( f(1) \) and \( f(2) \)

If \( f(1) \) and \( f(2) \) have different signs, it means that the function crosses the x-axis between 1 and 2 (since continuous functions cannot 'jump' across the x-axis without touching it), and the IVT applies.

Step 3: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem, if a function \( f \) is continuous on the interval [a, b] and k is any number between \( f(a) \) and \( f(b) \), then there exists a number c in the interval (a, b) such that \( f(c) = k \).

Key Concepts

Understanding Polynomial Zeros

The concept of polynomial zeros is central to understanding many areas of mathematics, particularly algebra and calculus. Simply put, a zero of a polynomial is any value of the variable, often denoted as x, at which the polynomial evaluates to zero. In other terms, if you plug in this value into the polynomial equation, the result is zero.

In the context of the exercise involving the polynomial equation f(x) = x^{3} - x - 1, finding its zeros is synonymous with finding those values of x for which the polynomial produces an output of zero. These points are of special interest because, graphically, they are where the curve of the polynomial intersects the X-axis.

The method of applying the Intermediate Value Theorem (IVT) to locate a polynomial zero is formidable. It relies on the understanding that if the function changes sign over an interval (from positive to negative or vice versa), there must be at least one point in this interval where the function equals zero. This sign change indicates a crossing over the X-axis, thus revealing the presence of a zero.

Continuity of Functions

The continuity of functions is an essential concept when discussing polynomial zeros, particularly when applying the Intermediate Value Theorem. A function is said to be continuous at a point if a small change in x results in a small change in the function's value f(x). More formally, a function f(x) is continuous on an interval if it is continuous at every point within that interval.

For polynomials, continuity is inherent. They are examples of functions that are continuous everywhere—in other words, on the entire set of real numbers. This property simplifies many problems in calculus because it guarantees that polynomials do not have breaks, jumps, or points of discontinuity.

When we speak of polynomials in relation to the IVT, their continuity guarantees that values transition smoothly from one point to another without any gaps. Therefore, if a polynomial changes signs over an interval, we are assured by its continuity that it has crossed the X-axis at some point in the interval.

Real Zeros of Polynomials

When we think about real zeros of polynomials, we are focusing on those zeros of the polynomial that are real numbers. These are distinct from complex zeros, which include imaginary numbers. Real zeros can be rational or irrational numbers and they represent intersections of the polynomial graph with the X-axis.

In practice, finding the real zeros of a polynomial can be done through various methods such as factoring, using the Rational Root Theorem, synthetic division, or graphing. However, when the roots are not easily identifiable, the Intermediate Value Theorem provides a powerful way to prove their existence, as seen in the given exercise, where it confirms a zero between 1 and 2 for the polynomial f(x) = x^{3} - x - 1.

By evaluating the function at these points and noting a change in sign, the theorem's conditions are satisfied, guaranteeing at least one real zero within that interval. This foundation is critical for moving on to more advanced calculus concepts, like finding exact values of zeros or determining the behavior of functions.