Problem 40 - College Algebra

Question

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

By substituting the values into the given equation, and because the value at 2 is positive and the value at 3 is negative, we can conclude using the Intermediate Value Theorem that the function indeed crosses the x-axis between \(x = 2\) and \(x = 3\). Therefore, there is at least one real zero in that interval.

Step 1: Calculate f(2) and f(3)

Determine the value of the function at the given points. Substitute \(x = 2\) and \(x = 3\) into the function.

Step 2: Check the Signs

If the result of f(2) is positive and the result of f(3) is negative (or vice versa), this means the function crosses the x-axis somewhere between 2 and 3 according to the Intermediate Value Theorem.

Step 3: Application of the Intermediate Value Theorem

As per the theorem, if f(a) and f(b) have different signs and the function is continuous on the interval [a,b], then it must cross the x-axis, i.e., there exists at least one c in (a, b) such that f(c) = 0.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables with constant coefficients. They can appear as linear, quadratic, cubic, or higher-degree forms. A basic layout of a polynomial is: \[f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\] where

• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants called coefficients.
• \(n\) is a non-negative integer representing the highest power of \(x\), known as the degree of the polynomial.

These functions are significant in numerous areas due to their simplicity and ability to approximate more complex functions. The example function in our exercise is a cubic polynomial, which means it has a degree of 3, with the highest term being \(3x^3\). Understanding polynomial functions helps prepare us for investigating other mathematical phenomena and real-world applications, from data modeling to physics. The behavior of these functions often impacts many fields, including engineering and economics.
They can be graphed easily, with one main feature being their smooth, curving lines without breaks.

Real Zeros

Real zeros of polynomial functions are the values of \(x\) where the function evaluates to zero, that is, \(f(x) = 0\). Finding these zeros can be essential, particularly when trying to understand the roots of equations in algebra and calculus. They essentially represent the x-intercepts of the graph of the function.

• They can be calculated exactly or approximately using various algebraic and numerical methods.
• Knowing about the existence of real zeros is vital for determining where a polynomial changes sign, which has numerous practical applications such as finding critical points.

In our exercise, we are asked to locate a real zero using the Intermediate Value Theorem (IVT) between two integers. This theorem is particularly helpful for continuous functions like polynomials, because it allows us to find real zeros without precisely calculating their value.
For our given polynomial, checking specific points such as 2 and 3, and finding a sign change, indicates a real zero within this interval.

Continuous Functions

Understanding continuous functions is crucial in applying the Intermediate Value Theorem. A function is continuous on an interval if there are no breaks, jumps, or holes in the graph within that interval.

• One fundamental property of continuous functions is that they can be drawn without lifting the pen from the paper along an interval.
• Polynomials are inherently continuous everywhere within their domain, which is all real numbers.

The continuity of functions is essential for many mathematical theorems and analyses. For example, it helps in analyzing behaviors of functions and predicting trends.
The Intermediate Value Theorem exploits this property of continuity. For our polynomial \(3x^3 - 8x^2 + x + 2\), we can state confidently that it is continuous because all polynomial functions are continuous. This property assures us that when the function changes signs across an interval of interest, it must have at least one real zero in that interval. This continuity, paired with the sign change between the calculated points of \(f(2)\) and \(f(3)\), allows the use of the IVT to confirm the presence of a real zero between these values.