Problem 34 - Algebra and Trigonometry

Question

In Exercises \(33-40,\) use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=x^{3}-4 x^{2}+2 ; \text { between } 0 \text { and } 1$$

Step-by-Step Solution

Answer

By applying the Intermediate Value Theorem to the function \(f(x)=x^{3}-4 x^{2}+2\), we can confirm that there is at least one real zero in the interval between \(x = 0\) and \(x = 1\).

Step 1: Compute the Function Values at the Endpoints of the Interval

First, evaluate the function \(f(x) = x^3 - 4x^2 + 2\) at the endpoints of the interval [0,1]. These values are:\n\n\(f(0) = (0)^3 - 4(0)^2 + 2 = 2,\)\n\(f(1) = (1)^3 - 4(1)^2 + 2 = -1.\)

Step 2: Apply the Intermediate Value Theorem

By the Intermediate Value Theorem, if a function is continuous on the interval [a,b], and \(k\) is any value between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in the open interval (a, b) such that \(f(c) = k\). Here, \(f(0) = 2\) and \(f(1) = -1\). As 0 is between these two values, we can say that there exists a \(c\) between [0,1] where \(f(c) = 0, i.e., a real zero exists in the interval.\)

Key Concepts

Real Zero of Polynomial

A real zero of a polynomial is a solution to the equation of the polynomial set equal to zero, which also belongs to the set of real numbers. In simpler terms, if you have a polynomial function such as f(x) = x^3 - 4x^2 + 2, finding a real zero means looking for a value of x, which we'll call c, that makes the function's value zero (f(c) = 0).

This concept is crucial in algebra because it represents the points where the graph of the polynomial intersects the x-axis. For instance, our function f(x) intersects the x-axis at some point between the integers 0 and 1, which we can determine using the Intermediate Value Theorem. By finding the real zeros of a polynomial, we're essentially solving the polynomial equation, and understanding how many such zeros exist helps us sketch the graph of the polynomial.

Continuity of Function

The concept of continuity of a function is fundamental in calculus and is intimately related to the behaviour of functions. A function is said to be continuous on an interval if there are no breaks, jumps, or holes in the graph over that interval. In formal terms, a function f is continuous at a point x = a if the limit of f(x) as x approaches a is equal to f(a).

Polynomial functions, like f(x) = x^3 - 4x^2 + 2, are always continuous everywhere, which means they're continuous over any interval you can think of. This is a property that comes in handy when we're trying to apply the Intermediate Value Theorem to find the real zeros of a polynomial, because continuity is a prerequisite for this theorem to be valid.

Polynomial Function

Characteristics of Polynomial Functions

A polynomial function is a mathematical expression that involves sums of powers of a variable, typically x, with non-negative integer exponents, and each term has a coefficient, which could be a real number. For example, the polynomial f(x) = x^3 - 4x^2 + 2 is a cubic polynomial, meaning its highest power of x is three.

Polynomials are smooth, continuous, and have well-defined properties, including their degree (related to the highest power), leading coefficient (the coefficient of the highest power), and roots or zeros (the values of x that make the polynomial equal to zero). They are very predictable in their behavior, making them some of the most studied objects in algebra and calculus.

Graphical Representation

Graphs of polynomial functions are always continuous curves without any breaks or sharp 'corners.' The degree of the polynomial gives us a clue about the number of peaks and valleys it might have, and hence the number of potential real zeros. The graph of our cubic polynomial could cross the x-axis up to three times, indicating up to three real zeros.

Finding Roots of Equations

The process of finding roots of equations essentially involves identifying values that satisfy the equation, meaning when those values are substituted into the equation, it holds true. Consider a general polynomial equation f(x) = 0. The solutions to this equation are called roots or zeros of the polynomial.

To find these roots, various methods are used depending on the type of equation. These methods range from graphing to using algebraic techniques such as factoring, applying the quadratic formula for second-degree polynomials, or more advanced methods like synthetic division and Newton's method for higher degrees.

An important aspect of finding roots is the use of theorems such as the Intermediate Value Theorem, which doesn’t necessarily give the exact roots but confirms their existence within a certain interval. For a function that is continuous on the interval [a, b], if f(a) and f(b) have opposites signs (one is positive and the other negative), then there's at least one real root between a and b.