Problem 35 - College Algebra

Question

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

Step-by-Step Solution

Answer

The Intermediate Value Theorem ensures there is a real root between -1 and 0 for the function f(x)=2x^4 - 4x^2 + 1.

Step 1: Plug in the interval endpoints in the function

Find the function values at -1 and 0. These are f(-1) = 2*(-1)^4 - 4*(-1)^2 + 1 = 2 - 4 + 1 = -1 and f(0) = 2*0^4 - 4*0^2 + 1 = 1.

Step 2: Check the signs of f(a) and f(b)

In this case, the endpoints are -1 and 0 with f(-1) = -1 and f(0) = 1. Now, since f(-1) and f(0) have different signs, the conditions of the Intermediate Value Theorem are met and it implies that there must be a number c in the interval (-1, 0) such that f(c) = 0.

Step 3: Conclude the solution

Since the function f(x) meets the conditions of the Intermediate Value Theorem on the interval [-1, 0], there is at least one real root of the equation f(x) = 0 between -1 and 0.

Key Concepts

Real Zero of a Polynomial

In mathematics, a real zero of a polynomial is a solution to the equation formed when the polynomial is set equal to zero, specifically a solution that is a real number. To find a real zero, we are essentially searching for the roots or x-intercepts of the polynomial function on a graph, which are the points where the graph crosses or touches the x-axis. For instance, in the exercise provided, we are looking to prove that there's at least one real zero for the function f(x) = 2x^4 - 4x^2 + 1 between the integers -1 and 0. This means we expect to find a value, let's name it c, within this interval where the polynomial equals zero. The proof uses the Intermediate Value Theorem, which guarantees the existence of this zero under specific conditions.

Polynomial Function

A polynomial function is a mathematical expression consisting of variables (also known as indeterminates), coefficients, and non-negative integer exponents of those variables. It's typically written in the form f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where n is a non-negative integer and each a represents a coefficient that can be a real number. A key feature of polynomial functions is their smooth and continuous nature, which allows us to use tools like the Intermediate Value Theorem to deduce properties about their zeros. In our exercise, the polynomial function given is of the fourth degree, suggesting it could potentially have up to four real zeros.

Roots of Equations

The term roots of equations refers to solutions of equations that set a particular function to zero. For polynomial equations, these roots are also called the zeros of the polynomial. To find these roots, various methods can be used, such as factoring, using the quadratic formula, or applying numerical techniques. The roots have significant implications in graphing the polynomial function, as they represent the points at which the graph intersects the x-axis. In simpler terms, the roots tell us where a function achieves a value of zero. For example, the polynomial in our exercise, f(x), will have its roots at the values of x for which f(x) = 0.

Continuity

The concept of continuity in mathematics describes a function that is unbroken or uninterrupted. A function is considered continuous on an interval if it has no holes, jumps, or breaks in that interval. The importance of continuity in the context of the Intermediate Value Theorem cannot be overstated, as the theorem requires the function to be continuous on the interval being considered. Continuity guarantees that for every value between f(a) and f(b), there is a corresponding c such that f(c) takes on that value. In our exercise, the polynomial function being continuous means that it flows smoothly without disruption between the interval [-1, 0], fulfilling a crucial condition for applying the theorem.