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A real zero of a polynomial is any real number that makes the polynomial equal to zero when substituted for the variable in the polynomial. In simpler terms, if you have a polynomial function, like the given example \(f(x)=x^{3}+x^{2}-2x+1\), and you're looking for its real zeros, you are searching for the x-values at which the graph of the function crosses or touches the x-axis.

To find a real zero between two numbers, you can use the Intermediate Value Theorem. This theorem states that if the polynomial function changes signs between two values, then at least one real zero must exist between those values. The reasoning behind this is grounded in the continuous nature of polynomial functions – they don't jump suddenly from one value to another without passing through all the values in between.

Evaluating a polynomial function involves replacing the variable in the polynomial with a specific number and performing the arithmetic to simplify the expression to a single numerical value. For instance, when you evaluate the function \(f(x)=x^{3}+x^{2}-2x+1\) at \(x=-3\) and \(x=-2\), you're essentially calculating the y-values at those points.

This process can be seen in the textbook exercise example, by substituting \(x=-3\) and \(x=-2\) into the function to get \(f(-3)\) and \(f(-2)\) respectively. This step is crucial to apply the Intermediate Value Theorem as it helps to determine if a change in the sign of function values occurs over the interval between the two x-values.

The signs of function values refer to whether the function's output is positive or negative for specific inputs. In the context of the Intermediate Value Theorem and finding real zeros, the signs are particularly important. They can give us a strong indicator of whether the function crosses the x-axis within a certain interval.

A change in sign, as mentioned in the textbook solution, occurs when the polynomial function's value moves from positive to negative or vice versa. If you have computed the values at the endpoints of an interval and they have opposite signs, then the function must have crossed the x-axis within that interval, implying the existence of at least one real zero.

The root existence in an interval concept is a direct application of the Intermediate Value Theorem. This theorem guarantees that if a continuous function, like a polynomial, switches signs over an interval, then there is at least one root (real zero) within that interval.

In practice, you'd evaluate the polynomial at both ends of the interval. For the function \(f(x)=x^{3}+x^{2}-2x+1\) given in the problem, we compute \(f(-3)\) and \(f(-2)\). If one value is positive and the other is negative, we've identified a sign change and thus confirmed a root exists between \(x=-3\) and \(x=-2\). It's this logic that provides a solid foundation for understanding the behavior of polynomial functions within a specified interval.