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When evaluating functions, the substitution method is an essential tool for replacing the independent variable with specified values. This process involves taking the given variable that is set equal to a number or another expression, and inserting it in place of the independent variable throughout the entire function.

For instance, if a function is written as \( h(t) = t^2 - 2t \), and we are asked to evaluate \( h(2) \), we substitute \( t \) with the number 2 in the function. The substitution is performed by replacing all instances of \( t \) with 2 and then simplifying to find the value of the function at that point. This technique is extremely useful because it allows us to determine the output of a function for any input we choose to plug in.

Polynomial functions are mathematical expressions that involve a sum of powers in one or more variables, with non-negative integer exponents and coefficients that can be real or complex numbers. These functions are incredibly versatile and appear frequently across various fields of mathematics and science.

A polynomial function like \( h(t) = t^2 - 2t \) is a relatively simple example, with terms that include a squared variable \( t^2 \), and a first-degree term \( -2t \). When working with polynomials, we often evaluate them using the substitution method as it is a systematic approach to determine their values at specific points. A polynomial's degree, which in this case is 2, reflecting the highest exponent, determines its basic shape and properties. For instance, second-degree polynomials commonly represent parabolic graphs that can open upwards or downwards.

Simplifying expressions is a fundamental process in algebra that involves reducing an expression to its most basic form without changing its value. This usually includes combining like terms, expanding multiplications, and applying arithmetic operations.

Whenever we evaluate a polynomial function after the substitution, as with \( h(x + 2) = (x + 2)^2 - 2(x + 2) \), we need to simplify the resulting expression. We first expand the squared term using the distributive property (also known as the FOIL method) to obtain individual terms. After expanding, we combine like terms, which are terms that have the same variable raised to the same power. For example, from the expression above after expanding and combining like terms, we would simplify \( h(x + 2) \) to get \( x^2 + 2x \), giving us a clean, simplified expression that conveys the same information in a more comprehensible way.