Problem 34 - College Algebra

Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{3}-4 x^{2}+2 ; \text { between } 0 \text { and } 1 $$

Step-by-Step Solution

Answer

By using the Intermediate Value Theorem, we can show that the polynomial \( f(x)=x^{3}-4 x^{2}+2 \) has at least one real zero in the interval between 0 and 1.

Step 1: Evaluate the Polynomial at Endpoints

Evaluate the function \( f(x)=x^{3}-4 x^{2}+2 \) at \( x=0 \) and \( x=1 \) to get \( f(0) \) and \( f(1) \). By substituting \( x = 0 \) into the polynomial, we get \( f(0) = 2 \). Likewise, substituting \( x = 1 \) we get, \( f(1) = -1 \)

Step 2: Apply Intermediate Value Theorem

Since \( f(0) = 2 \) and \( f(1) = -1 \), and 0 is between these two function values, it is confirmed that there is at least one number \( c \) in the interval [0, 1] such that \( f(c) = 0 \). This is ascertained by the Intermediate Value Theorem which posits that since \( f(x) \) is a polynomial, it is continuous, and thus, by IVT, since 0 is included in the interval between \( f(0) \) and \( f(1) \), there must be \( c \), \( 0 < c < 1 \), such that \( f(c) = 0 \).

Step 3: Conclusion

It can be concluded that by the Intermediate Value Theorem, there is a real zero between the given integers 0 and 1 for the polynomial \( f(x)=x^{3}-4 x^{2}+2 \).

Key Concepts

Real Zero of Polynomial

Ranging from simple linear equations to complex expressions, polynomials play a critical role in various fields of mathematics and science. One question that often arises is how to determine if a polynomial has a real zero, or solution. A real zero of a polynomial is where the graph of the polynomial crosses or touches the x-axis, which corresponds to a value of x for which the polynomial equals zero.

In the provided exercise, the task was to use the Intermediate Value Theorem to establish that there is at least one real zero for the polynomial function f(x) = x^3 - 4x^2 + 2 between the integers 0 and 1. In simpler terms, students are challenged to prove that the graph crosses the x-axis within this interval. By evaluating the polynomial at the endpoints of the interval and applying the Intermediate Value Theorem, a conclusive demonstration of a real zero's presence was provided. This approach is a powerful tool for mathematicians, as it can assure them of the existence of a real zero without necessarily finding its exact value.

Evaluating Polynomials

Understanding how to evaluate a polynomial is a fundamental skill that every math student needs to develop. Evaluating polynomials involves substituting a number for the variable and calculating the resulting value. The process is straightforward: for each term in the polynomial, multiply the coefficient by the value raised to the power indicated by the exponent, and then sum all the terms together.

For example, in the exercise, the polynomial f(x) = x^3 - 4x^2 + 2 was evaluated at the endpoints of the given interval, 0 and 1. Such evaluation is integral to applying the Intermediate Value Theorem, as it gives us the necessary function values to establish a change in sign (from positive to negative or vice versa), which indicates a zero within the interval. It is through this method that students are able to connect the abstract concept of a polynomial with a tangible function value that can be analyzed within the context of mathematical theorems.

Polynomial Continuity

A core aspect of polynomials that enables the use of the Intermediate Value Theorem is their continuity. A function is said to be continuous on an interval if it has no breaks, jumps, or holes on that interval. The beauty of polynomial functions is their inherent continuity — they are continuous for all real numbers, meaning they will not suddenly leap or become undefined, which is crucial for the reasoning used in the Intermediate Value Theorem (IVT).

In our exercise context, IVT states that for a continuous function like a polynomial, if you have two points on either side of the horizontal axis (y=0), there must be at least one point between those two where the function crosses the axis. The smooth, unbroken nature of polynomials makes them a perfect candidate for the IVT, and in educational settings, polynomials often serve as the first introduction to this powerful theorem. Grasping the concept of continuity in relation to polynomials allows students to understand more complex applications of continuity in calculus and beyond.