91Ó°ÊÓ

A polynomial function is an expression that involves only the arithmetic operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are significant in mathematics due to their versatility in modeling various real-world situations and their profound properties such as symmetry. In this exercise, the given function, \( k(x) = 5x^3 + \frac{5}{x^3} - x - \frac{1}{x} \), combines polynomial terms and terms with reciprocal powers, highlighting that polynomial-type expressions can extend to functions involving reciprocal powers while still retaining specific algebraic properties. Polynomial functions are generally continuous and differentiable, which makes them crucial for both theoretical mathematics and practical applications.

Function symmetry pertains to a situation where a function exhibits mirrored properties. The symmetry displayed in the problem reflects what is known as reciprocal symmetry. Specifically, this dictates that for the function \( k(x) \), swapping \( x \) with \( \frac{1}{x} \) yields the same functional output without affecting the overall expression. In simpler terms, for every value of \( x \), the transformation \( k(\frac{1}{x}) \) results in the same outcome as the original function \( k(x) \). This particular kind of symmetry shows a relationship in the function that is equivalent across different representations. Identifying symmetry helps simplify complex evaluations of functions and enables deeper insights into their structural properties.

Algebraic manipulation involves rearranging and simplifying expressions to find equivalent formulations or solve equations. In the exercise at hand, algebraic manipulation is key to proving \( k(x) = k(\frac{1}{x}) \). By substituting \( \frac{1}{x} \) into the function and applying rules of exponents, we can simplify the function step by step:

• Recognize that raising reciprocal to the power results in obtaining reciprocal terms: \( \left(\frac{1}{x}\right)^3 = \frac{1}{x^3} \).
• Simplify the coefficient multiplications: \( 5\left(\frac{1}{x}\right)^3 = \frac{5}{x^3} \) and \( \frac{5}{\left(\frac{1}{x}\right)^3} = 5x^3 \).
• Keep other terms consistent, ensuring the algebraic transformation remains valid without altering the sign or the position: \(-\frac{1}{x}\) and \(-x\) stay the same.

Ultimately, through precise manipulation, the expressions \( k(x) \) and \( k\left(\frac{1}{x}\right) \) are both effectively simplified to be equal, evidencing the symmetry using straightforward algebra.