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A polynomial function is a mathematical expression consisting of variables and coefficients, where the variables are raised to whole-number exponents. These functions can be used to model various types of problems in algebra and applied mathematics. In the given exercise, the polynomial function is represented as:

• \(f(x) = x^4 + 6x^3 - 18x^2\)

A polynomial function is defined by its degree, which is the largest exponent of its variable. For example, in our exercise, the highest power of \(x\) is 4, making it a fourth-degree polynomial. Polynomials can often describe curves and can have one or more zeros, depending on their degree. This becomes crucial when applying concepts like the Intermediate Value Theorem to find these zeros.

Real zeros of a polynomial function are the x-values where the function equals zero. These are also known as the roots of the polynomial. Real zeros are important as they indicate where the graph of the polynomial crosses or touches the x-axis. In our exercise:

• We want to find a real zero of \(f(x) = x^4 + 6x^3 - 18x^2\) between the integers 2 and 3.

Real zeros can be found using different methods, including factoring, graphing, or applying the Intermediate Value Theorem. Finding a real zero helps in understanding the behavior of the polynomial across different intervals and is often a crucial step in solving or simplifying algebraic equations. Understanding where the polynomial is zero assists in graphing the function effectively.

Root finding involves identifying the x-value where a polynomial function equals zero. A powerful method to find roots in a specific interval is the Intermediate Value Theorem (IVT). This theorem states:

• If a continuous function changes signs over an interval, then it has at least one real zero in that interval.

In the given exercise, we evaluated \(f(x)\) at two points:

• \(f(2)\) and \(f(3)\)

If \(f(2)\) and \(f(3)\) have different signs (one positive and one negative), then according to the IVT, \(f(x)\) must have at least one real zero between x=2 and x=3. This theorem is especially handy for complex polynomials where factoring is difficult, providing a straightforward logical approach to determine the existence of roots.

Function evaluation entails substituting specific values into a function to compute the output. This is a basic yet powerful tool in mathematics used to analyze and understand the behavior of functions. In the context of our exercise:

• We evaluated the function \(f(x) = x^4 + 6x^3 - 18x^2\) at \(x = 2\) and \(x = 3\) to aid in applying the Intermediate Value Theorem.

Evaluating a function at different points is crucial, as it helps us understand the trend or behavior of the function across a particular range. When utilizing the Intermediate Value Theorem, it is essential to evaluate the function at the boundaries of the given interval to check whether the signs of the outputs differ. This systematic approach not only confirms the presence of zeros but also assists in determining intervals where the function changes direction.