Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

By the Intermediate Value Theorem, the polynomial \( f(x)=2 x^{4}-4 x^{2}+1 \) has at least one real zero between \( -1 \) and \( 0 \)

Step 1: Calculate \( f(-1) \)

First, substitute \( x = -1 \) into the function \( f(x)=2 x^{4}-4 x^{2}+1 \). This yields \( f(-1) = 2*(-1)^{4}-4*(-1)^{2}+1 = 2-4+1= -1 \)

Step 2: Calculate \( f(0) \)

Next, substitute \( x = 0 \) into the function \( f(x)=2 x^{4}-4 x^{2}+1 \). This yields \( f(0) = 2*0^{4}-4*0^{2}+1 =1 \)

Step 3: Apply the Intermediate Value Theorem

Since \( f(-1) = -1 \) and \( f(0) = 1 \), there is a sign change between these values. According to the IVT, this implies that the function has at least one real zero between \( -1 \) and \( 0 \)

Key Concepts

Polynomial Zeros

When delving into the realm of algebra, one concept that frequently arises is that of 'polynomial zeros'. But what exactly are polynomial zeros? In simple terms, a 'zero' of a polynomial is a value for the variable, commonly referred to as 'x', which makes the polynomial equal to zero. Suppose we have a polynomial function such as f(x) = x^2 - 5x + 6. The zeros of this polynomial are the solutions to the equation f(x) = 0.

In the original exercise, f(x) = 2x^4 - 4x^2 + 1, finding the real zeros between -1 and 0 involves using the Intermediate Value Theorem (IVT). The IVT tells us that if we have a continuous function--and polynomials are always continuous--and there is a change in sign between two values, then there must be at least one zero in that interval. By calculating f(-1) = -1 and f(0) = 1, we find a change in sign which confirms the presence of a zero between -1 and 0.

Zeros are the points where the graph of the polynomial crosses or touches the x-axis. They are essential in understanding the behavior of polynomials, as they can give us valuable information about factors, roots, and x-intercepts of the function.

Real Zeros of Polynomials

When students learn about the 'real zeros of polynomials', they're uncovering where a polynomial function touches or crosses the x-axis on a graph, which corresponds to the real number solutions of the equation f(x) = 0. While a polynomial might have complex zeros (involving imaginary numbers), real zeros are the x-values we can plot on a standard two-dimensional graph.

To determine the real zeros of a polynomial, various methods are used, including factoring, synthetic division, and applying theorems such as the Intermediate Value Theorem. In our exercise, by showing that f(x) changes sign between -1 and 0, the Intermediate Value Theorem assures us of the existence of at least one real zero in that interval without actually having to solve the equation.

This is incredibly useful because sometimes finding the explicit zeros algebraically can be either very cumbersome or even impractical with the mathematical tools at hand. Theorems like IVT can cut down the work and show us where to look for these solutions or confirm their existence.

Calculating Polynomial Functions

While the concept of 'calculating polynomial functions' may sound daunting, it's really a systematic method that allows us to evaluate the function at specific values of 'x'. This is a fundamental skill in algebra as it enables students to analyze and graph polynomial functions effectively.

In the context of our exercise, we calculate the polynomial function f(x) = 2x^4 - 4x^2 + 1 at two specific points: x = -1 and x = 0. By substituting these values into the polynomial, we obtain f(-1) = -1 and f(0) = 1 respectively. These calculations are crucial because they allow us to apply the Intermediate Value Theorem, leading to the conclusion that there's at least one real zero between -1 and 0.

To evaluate a polynomial function, you'll usually start by replacing the variable 'x' with the given number and then follow the operations indicated by the function. It's important to mind the order of operations (PEMDAS—parentheses, exponents, multiplication and division, and addition and subtraction) to accurately calculate the value of the function. Mastery of this process is key to advancing in understanding and working with polynomials.