Definition: The term "algebraically" refers to a method or process by which one constructs formulas or equations that can be used in algebra, calculus, and other mathematical fields. Algebraic expressions are written using letters such as x, y, z, etc., and are manipulated with variables such as a, b, c, d, etc. These variables represent unknown values that need to be determined. The word "algebraically" comes from the Greek phrase "algebras," which means "mathematical." In this context, it suggests using mathematical principles to solve problems or find solutions in algebra and calculus. Here is a detailed definition of "algebraically": - Algebraically: The method of manipulating symbols and expressions using letters as variables, where these are used to represent unknown numbers or values that need to be determined. Example: Suppose we want to create an equation for a circle. We can do this by algebraically defining the variables x and y, then setting them equal to one another (x + y = 0) to find the value of the unknown variable "y." This is how algebraically solving problems works. - Method: The use of letters as symbols to represent unknown quantities or values in equations is known as algebra. The process can be applied to a wide variety of mathematical operations, including simplification and manipulation of expressions. Example: In the expression (2x^3 + 7x - 5) / (x - 1), we use algebraic methods to rewrite it as: (2x^2
x + 2x
x) / (x - 1) This step is known as "dividing" or "simplifying," and can be seen in the steps where we move one term inside the parentheses, then multiply by the reciprocal of the second term. - Properties: The algebraic method relies heavily on properties of variables and operations within mathematical expressions. For example, when dividing or multiplying two polynomials, the first polynomial is moved outside the parentheses to match the second polynomial's expression. In summary, "algebraically" refers to a systematic approach in which letters are used as symbols for unknown quantities and values to be determined through manipulation of mathematical expressions. This method enables solving problems involving algebraic equations or operations where variables are involved.

algebraically