Final answer:
The terms in the expression are named coefficients and these are -81, 3, -12, 9, and 1 for the implied term x. Understanding the coefficients and their role is crucial for operations like series expansions or when dealing with exponentials.
Step-by-step explanation:
The name of the terms in the expression xâ¹-81xâ¸+3x⁶-12x⁵+9x²+x are coefficients. Each term in a polynomial is made up of a coefficient and a variable raised to an exponent. The coefficient is the numerical part of the term, while the variable part defines the term's degree based on its exponent. In the expression provided, the coefficients are -81, 3, -12, 9, and 1 (implied for x), and the variables are x raised to various exponents.
When considering series expansions or cubing of exponentials, it's important to understand how coefficients and exponents are managed. For instance, cubing an expression like (2x) involves raising both the numerical coefficient (2) and the variable term with its exponent by the power of 3, leading to 2³*x³¹. This can be seen in series expansions where each term is affected by the operations applied to x.