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The difference quotient is a fundamental tool in calculus used to find the derivative of a function at a particular point. It essentially measures the average rate of change of the function over a small interval. We express it mathematically as \( \frac{f(x+h) - f(x)}{h} \). This formula evaluates how the output of a function changes as the input is increased by a small amount \(h\). It serves as the precursor to the derivative, which is the rate of change at a single instant.
In the exercise, the function \(F(t) = 4t^3\) requires us to find the simplified form of \([F(a+h) - F(a)] / h\). By substituting and simplifying, one determines how the function \(F(t)\) behaves as its variable \(t\) slightly increases from \(a\) to \(a+h\).

• The numerator \([F(a+h) - F(a)]\) represents the change in the function's value between \(a\) and \(a+h\).
• Dividing by \(h\) gives us the average rate of change over that interval.
• As \(h\) approaches zero, the difference quotient gives us the derivative of the function.

The binomial theorem provides a method to expand expressions that are raised to a power, especially of the form \((a + b)^n\). An expansion using the binomial theorem is written as:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This formula allows us to expand polynomial expressions easily, which is crucial in calculus and algebra.
In our problem, the binomial theorem is applied to expand \((a+h)^3\). It shows how we can break down complex expressions into simpler terms:

• \(a^3\) - The term when only \(a\) appears from the binomial term.
• \(3a^2h\) - The term accounting the combination of \(a\) appearing twice and \(h\) appearing once.
• \(3ah^2\) - Illustrates \(a\) once and \(h\) twice.
• \(h^3\) - Represents \(h\) raised to the third power exclusively.

These expansions help to systematically simplify and solve equations involving polynomial expressions.

Polynomial functions are expressions involving variables raised to whole number powers. They can be simple, like a quadratic \(ax^2 + bx + c\), or more complex like the cubic function found in our exercise, \(4t^3\). They are characterized by their smooth and continuous curves.
Polynomials are highly useful in differential calculus because their derivatives are straightforward to calculate. In fact, the power rule - which simplifies the differentiation of polynomials - states that the derivative of \(x^n\) is \(nx^{n-1}\).
In the problem's context, the polynomial \(4t^3\) involves:

• The highest power is 3, indicating a cubic polynomial.
• As per binomial expansion and simplification, the function transforms and simplifies in its difference quotient form.
• Polynomials remain a cornerstone in calculus due to their predictable behavior and wide application.