Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Calculation at the endpoints

Evaluate the function \(f(x)=x^{3}+x^{2}-2x+1\) at points -3 and -2. The function is \(f(-3)=(-3)^{3}+(-3)^{2}-2(-3)+1= -27+9+6+1=-11\) and \(f(-2)=(-2)^{3}+(-2)^{2}-2(-2)+1= -8+4+4+1=1\).

Step 2: Compare the signs of the function values

Now, note that \(f(-3)=-11<0\) and \(f(-2)=1>0\). This means, the function moves from a negative to a positive value as x moves from -3 to -2.

Step 3: Apply the Intermediate Value Theorem

As the function \(f(x)=x^{3}+x^{2}-2x+1\) is a polynomial function, it's necessarily continuous everywhere in its domain, including between -3 and -2. As it moves from below zero to above zero over this interval, the Intermediate Value Theorem (IVT) ensures there must be a point within the interval (-3,-2) at which the function is zero.

Key Concepts

Real Zeros of Polynomials

The quest to locate the real zeros of polynomials is a core pursuit in algebra. Real zeros are the x-values where the polynomial evaluates to zero; graphically, these are the points where the curve crosses or touches the x-axis. The polynomial function presented in the exercise, f(x) = x^3 + x^2 - 2x + 1, is examined between the integers -3 and -2. Using the Intermediate Value Theorem (IVT) to show that a real zero exists between these two points hinges on understanding that a continuous function that changes signs over an interval (from positive to negative or vice versa) must cross the x-axis, which is to say, it must have a real zero in that interval.

In exploring this concept, it's vital to note that finding the exact value of the zero might require more sophisticated methods such as factoring, synthetic division, or numerical approximation techniques like the bisection method. Nevertheless, the IVT provides the guarantee that between -3 and -2 lies at least one real zero for the polynomial in question.

Continuity of Polynomial Functions

A pivotal property of polynomial functions is their continuity on the entire set of real numbers. Continuity, in a less technical sense, means that you can draw the graph of the function without lifting your pencil. The function in this exercise, f(x) = x^3 + x^2 - 2x + 1, like all polynomial functions, meets this criterion and hence can be studied using the Intermediate Value Theorem.

A deeper dive into this concept reveals that continuity guarantees no sudden jumps, breaks, or holes in the graph of the polynomial function. It implies that for any two points on the function's graph, every value between their outputs is also attained. This behavior is a prerequisite for the applicability of the IVT. When considering the problem at hand, once we know the function is continuous, we are assured that if there is a sign change on the outputs (as in the values of f(-3) and f(-2)), there must be at least one real zero in between.

Evaluating Polynomial Functions

Evaluating polynomial functions at specific points is an essential operation for understanding their behavior. In the given exercise, f(x) is evaluated at x = -3 and x = -2, yielding the function values f(-3) = -11 and f(-2) = 1, respectively. This step is crucial to determine the change in the function's signs over the chosen interval.

Through this evaluation, one can discern how the function transitions from one point to another, highlighting any sign changes. Polynomials being smooth and continuous curves without gaps, this evaluation is all the more predictable and useful. When the function value transitions from negative to positive or vice versa, as exhibited by f(-3) and f(-2), it implies that the function intersects the x-axis, hence the existence of a real zero in that interval. Mastery of polynomial function evaluation is not just useful for finding zeros; it also aids in sketching graphs, solving optimization problems, and analyzing function behavior across domains.