91Ó°ÊÓ

Polynomial functions, like the one in our exercise, play a key role in calculus and algebra. A polynomial function is a type of function that consists of terms that are non-negative integer powers of a variable. The function often takes the form of \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\] Here, each of the \(a_n\)s are coefficients, and \(x\) represents the variable. In this specific exercise, the polynomial function is \(f(x) = x^3 + 3x\). It consists of two terms. One term is \(x^3\), which is cubic, and the other is \(3x\), which is linear.

• The highest power of \(x\) in the polynomial \(f(x) = x^3 + 3x\) is 3, so it's a third-degree polynomial.
• The degree of a polynomial is determined by the highest exponent of \(x\).

Knowing how to handle and manipulate polynomial functions is essential for solving many calculus problems, including difference quotient calculations.

Function notation is a way to express and evaluate functions. It helps us substitute and evaluate functions easily with different variables or values. For example, the function \(f(x) = x^3 + 3x\) shows us that the output \(f(x)\) is generated when we apply the expression on the right to \(x\). When we use notation like \(f(x+h)\), this implies that we replace every \(x\) in our function with \(x+h\).

• To calculate \(f(x+h)\) for our polynomial, substitute every instance of \(x\) in \(f(x)\) with \(x+h\).
• This gives us \((x + h)^3 + 3(x + h)\).

Function notation is essential when handling complex expressions, such as in calculus, because it provides a concise way to express variable relationships.

Simplifying expressions is vital to make complex mathematical expressions more manageable. Simplification usually involves applying algebraic rules to collapse or combine terms to their simplest form. In our exercise, we start by expanding and simplifying the expression \(f(x+h) - f(x)\). To expand \((x+h)^3\), apply the binomial theorem to get \(x^3 + 3x^2h + 3xh^2 + h^3\). Then distribute \(3\) over \((x + h)\) to simplify further. Once simplified, the expression should be put over \(h\) according to the difference quotient formula: \(\frac{(x + h)^3 + 3(x + h) - (x^3 + 3x)}{h}\).

• Combine like terms to get \(3h^2 + 3h\) in the numerator.
• Cancel out \(h\) from the numerator and the denominator.
• This leaves us with the simplified result as \(3h + 3\).

The art of simplifying ensures expressions are easier to handle, solve, and understand.