Problem 40 - 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. $$f(x)=3 x^{3}-8 x^{2}+x+2 ; \text { between } 2 \text { and } 3$$

Step-by-Step Solution

Answer

Yes, according to the Intermediate Value Theorem, the function \(f(x)=3 x^{3}-8 x^{2}+x+2\) does have a real zero between the integers 2 and 3.

Step 1: Evaluating the function at the interval boundaries

First, let's find out the values of the function \(f(x)\) at \(x=2\) and \(x=3\). So substitute \(x=2\) and \(x=3\) into the function.

Step 2: Calculation of the function values

By substitution we have:\n \(f(2)= 3*(2)^{3}-8*(2)^{2}+2+2 = -4\)\n \(f(3)= 3*(3)^{3}-8*(3)^{2}+3+2 = 10\)\n As you can see, \(f(2)\) and \(f(3)\) have different signs, which means the function value changes from negative to positive as \(x\) increases from 2 to 3.

Step 3: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem, if the function is continuous at every point in some interval \([a, b]\) and \(d\) is between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in the interval \([a, b]\) such that \(f(c) = d\). Here, our function is a polynomial, and polynomials are always continuous. Therefore, according to the Intermediate Value Theorem, there is a point \(c\) between \(x = 2\) and \(x = 3\) where \(f(c) = 0\).

Key Concepts

Real Zeros of Polynomials

When we talk about 'real zeros of polynomials', we are referring to the values of x for which the polynomial function returns a value of zero. These zeros are also known as 'roots' of the polynomial. For example, consider the polynomial function given in the exercise, f(x) = 3x^3 - 8x^2 + x + 2. To find its real zeros, one must solve the equation f(x) = 0.

Polynomial functions may have multiple real zeros, but the number of real zeros is always less than or equal to the degree of the polynomial. In the provided example, since the polynomial is a cubic one (degree 3), there can be at most three real zeros. The exercise specifically targets the search for a zero between two given integers, 2 and 3. This is where we use methods like graphing, factoring, synthetic division, or as illustrated in the exercise, the Intermediate Value Theorem, to conclude the existence of at least one real zero in that interval.

Continuity of Polynomial Functions

Polynomial functions are a basic example of continuous functions. Continuity, in a layman's term, means that you can draw the graph of the function without lifting your pencil from the paper. This mathematical property is crucial for the Intermediate Value Theorem (IVT) to be applicable.

A function is said to be continuous on an interval if, at every point within the interval, the function has no holes, jumps, or vertical asymptotes. This is why the step by step solution of our exercise did not require checking for continuity: a polynomial function like f(x) = 3x^3 - 8x^2 + x + 2 is always continuous everywhere on the real number line. Hence, it satisfies the continuity requirement for the IVT, making it a reliable tool for finding real zeros.

Finding Roots of Equations

Finding roots of equations, particularly polynomial equations, is a fundamental task in algebra. The 'roots' or 'solutions' are the x-values where the function equals zero (its y-value). There are various methods to find the roots, such as factoring if the polynomial simplifies nicely, using the quadratic formula for second-degree polynomials, or employing numerical methods for complex or higher-degree ones.

The Intermediate Value Theorem provides a theoretical foundation to ensure the existence of a root within a certain interval, for continuous functions. If you calculate the function’s value at two points and find they have opposite signs, as shown in the exercise with f(2) = -4 and f(3) = 10, there must be at least one point - a root - where the function crosses the x-axis between these two points. To actually find the root, one might use numerical methods like the bisection method, Newton's method, or simply rely on graphing calculators or computer algorithms to approximate its value.