Problem 7 - College Algebra

Question

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=x^{3}-x-1 ;\) between 1 and 2

Step-by-Step Solution

Answer

The given polynomial \(f(x)=x^{3}-x-1\) has at least one real zero between 1 and 2. The approximate value of this zero, to the nearest tenth, is 1.5

Step 1: Calculate the function values at given intervals

Calculate the function \(f(x)=x^{3}-x-1\) at x=1 and x=2. We get \(f(1)= (1)^{3} -(1) -1= -1\) and \(f(2)= (2)^{3} -(2) -1=5.\)

Step 2: Apply Intermediate Value Theorem

According to the Intermediate Value Theorem, since \(f(1)\) is negative and \(f(2)\) is positive, and considering that the function \(f(x)\) is continuous, there must exist at least one root of \(f(x)\) between 1 and 2.

Step 3: Find Approximate Zero

Using the bisection method, the midpoint between 1 and 2 is \(1.5\) and \(f(1.5)= (1.5)^{3} -(1.5) -1 = 0.125\) which is closer to zero. The midpoint between 1.5 and 2 is \(1.75\) and \(f(1.75)= (1.75)^{3} -(1.75) -1 = 1.421875\) which is further from zero. Therefore, 1.5 is a better approximation.

Step 4: Check with a Graph

If applicable, verify this result by looking at the graph of the function and observing that the function does indeed cross the x-axis between 1 and 2. This verifies that 1.5 is indeed an approximation to the zero of the function. Please note that graphing is not always applicable in every exercise.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. They involve terms that have non-negative integer exponents. For example, the given function in the exercise is a polynomial function: \(f(x) = x^3 - x - 1\). This is a cubic polynomial because the highest exponent of the variable \(x\) is 3.

Polynomials can have one or more terms, and they often appear in the form:

• \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\) where \(a_n, a_{n-1}, ..., a_0\) are constants and \(a_n eq 0\).

Polynomials are important in mathematics because they can model a wide array of real-world situations.

They are also continuous and smooth, which means they do not have breaks or sharp turns in their graphs. This plays a key role when using the Intermediate Value Theorem to find real zeros.

Bisection Method

The bisection method is a numerical technique used to approximate the roots or zeros of continuous functions. It is particularly useful when you know that a function changes sign over an interval, which implies a zero exists by the Intermediate Value Theorem.

In the original exercise, we used the bisection method to find an approximation of the zero of \(f(x) = x^3 - x - 1\) between 1 and 2. Here's how it works:

• Start with two points \(a\) and \(b\) where the function changes sign (i.e., \(f(a)\) and \(f(b)\) have opposite signs).
• Calculate the midpoint, \((a+b)/2\), and determine if this point is a root or changes the sign of the function.
• If the midpoint is closer to zero or changes the function's sign, replace one of your interval's endpoints with this midpoint.
• Repeat this process to narrow down the interval until the approximate root is found to the desired accuracy.

This method is simple yet powerful for finding zeros when you want precision, and the function is continuous over the interval.

Real Zeros

Real zeros of a function are the x-values where the function equals zero. These points are where the graph of the function crosses or touches the x-axis.

In the exercise, we found that there is at least one real zero for the polynomial \(f(x) = x^3 - x - 1\) between 1 and 2. By using the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval, there must be a zero within that interval.

Identifying real zeros is essential in solving equations and understanding the behavior of polynomials. It is foundational in calculus and real analysis, where zero-crossings can indicate important characteristics of the function. Real zeros often provide crucial insights in fields such as physics, engineering, and economics, where such functions might describe real-world systems or phenomena.