91Ó°ÊÓ

Polynomial functions are mathematical expressions that involve variables raised to non-negative integer exponents. They are among the most common types of functions encountered in algebra. The function provided in the exercise, \(f(x) = 2x^2 - x + 8\), is a polynomial function of degree 2.

Understanding polynomial functions is important because they model many real-life situations, such as projectile motion and area calculations. They follow the general form: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a coefficient and \(n\) determines the degree. This specific function has:

• Two terms contributed by the \(x^2\) and \(x\) components.
• A constant term, which is 8 in this instance.

Understanding the structure of polynomial functions makes operations like addition, subtraction, and differentiation more manageable, as they can be performed term-by-term.

Function evaluation is the process of finding the output of a function for a given input. It involves substituting specific values for the variable in the function equation. Here, we find \(f(a+h)\) and \(f(a)\) to compute the difference quotient. Understanding how to evaluate functions is essential for calculations like these.

To evaluate \(f(x) = 2x^2 - x + 8\) at \(x = a + h\), substitute \(a + h\) in place of \(x\), resulting in \(f(a+h) = 2(a+h)^2 - (a+h) + 8\). Next, expand and simplify it. Similarly, for \(x = a\), the function evaluates to \(f(a) = 2a^2 - a + 8\).

Function evaluation helps in:

• Understanding how functions behave with different inputs.
• Facilitating the analysis of changes in functions over specific intervals or with increments like \(h\).
• Providing a means to work with complex functions by breaking them down into simpler sub-problems.

Algebraic simplification is the process of rewriting expressions in a simpler or more understandable form. In this exercise, it involves expanding expressions and combining like terms to simplify the difference quotient.

When simplifying the function \(f(a+h) = 2(a+h)^2 - (a+h) + 8\), start by expanding \((a+h)^2\) to obtain \(a^2 + 2ah + h^2\). Then distribute the 2, giving \(2a^2 + 4ah + 2h^2\).

• Combine these with \(-a - h + 8\)
• Simplified, this becomes \(2a^2 + 4ah + 2h^2 - a - h + 8\).

After evaluating \(f(a)\) to be \(2a^2 - a + 8\), compute the difference \(f(a+h) - f(a)\), and simplify to \(4ah + 2h^2 - h\).

Finally, dividing \(4ah + 2h^2 - h\) by \(h\) leads to the final simplified form, \(4a + 2h - 1\). Simplification makes it easier to interpret mathematical results by reducing complexity and highlighting the main features of the expressions.