Final answer:
A Polynomial Identity is a true statement for any values of the variables, and remains true when both sides are manipulated equally, such as by multiplication or division with brackets enclosing multiple terms.
Step-by-step explanation:
A Polynomial Identity is a mathematical statement that holds true for all values of the variables involved. When we say an identity is true, it implies that no matter how we manipulate the expression, as long as we do the same operation to both sides, the equality remains valid.
Multiplication or division by the same number on both sides of an equation is a common example of such a manipulation that maintains the truth of the equality. It is essential to enclose the side with more than one term in small brackets and then perform the multiplication or division operation so that it applies to every term within the brackets.
As an example, consider the manipulations of temperature conversion equations or ideal gas equations. These demonstrate how polynomials and monomials can be manipulated while still preserving fundamental truths about the equations.
In all these cases, each side of the equation can be seen as a balance, with changes to one side necessitating identical changes to the other to maintain balance or equality. This concept is a crucial foundation in algebra and helps us understand and solve equations effectively.