91Ó°ÊÓ

Polynomial functions are a cornerstone of algebra and calculus. They are expressions that consist of variables and coefficients, shaped by operations of addition, subtraction, multiplication, and exponentiation by a non-negative integer. For example, in the function \( f(x) = 5x^2 \), the term \( 5x^2 \) is a monomial, which is a single term that is itself a simple polynomial. Polynomials are defined by their degree, which is the highest power of the variable present in the expression. In \( f(x) = 5x^2 \), the degree is 2. Understanding the structure and behavior of polynomial functions is crucial for analyzing real-world phenomena, as they can be used to model diverse scenarios from physics to economics.

Evaluating a function involves substituting a given input value into the function to calculate the output. In the context of the exercise, we evaluate \( f(x+h) \) by substituting \( x+h \) into the function \( f(x) = 5x^2 \). By doing this, we are essentially determining the effect of a small change (\( h \)) on the variable \( x \). This process is vital in calculus, especially in concepts like limits and derivatives, where we study how functions behave as inputs change slightly. Here's how to evaluate:

• Identify the function and specify the new input.
• Replace the variable \( x \) with \( x+h \) in the function \( f(x) \).
• Perform algebraic operations to simplify the new expression.

In this way, function evaluation allows us to understand changes and predict outcomes in various fields.

Algebraic manipulation is a set of techniques used to simplify or rearrange expressions and equations. In the exercise, we perform algebraic manipulation by expanding and simplifying \( (x+h)^2 \), and then multiplying the result by 5.This involves several steps:

• Apply the binomial expansion formula: \( (a+b)^2 = a^2 + 2ab + b^2 \).
• In our example: replace \( a \) with \( x \) and \( b \) with \( h \). This gives \( x^2 + 2xh + h^2 \).
• Multiply each term by the coefficient present in the original function, here it's 5, leading to \( 5x^2 + 10xh + 5h^2 \).

These steps help streamline problems, making complex expressions easier to understand and solve. Mastery of algebraic manipulation is essential for tackling polynomial functions, simplifying expressions, and solving equations efficiently.