91Ó°ÊÓ

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the exercise \(\left(x^{2}-4\right)-\left(x^{2}-4 x\right)\), we have an example of a polynomial expression. The variables in this case are represented by \(x\) and \(x^2\), and the numbers -4 and +4 are constants. When dealing with algebraic expressions, it's essential to follow the rules of arithmetic operations, specifically how to handle the addition and subtraction of polynomials.

Just like in numerical subtraction, where you line up the numbers by their place value, algebraic subtraction aligns terms with the same variables and powers. This process often involves expanding brackets and ensuring that signs are applied correctly, so each term is accurately accounted for in the operation.

Combining like terms is a crucial step in simplifying algebraic expressions. In the context of subtraction, like terms have the same variable raised to the same power. For instance, \(x^2\) and \(x^2\) are like terms, meanwhile \(x^2\) and \(4x\) are not, since they involve different variables or powers.

In the given solution, \(x^2\) and \(\-x^2\) are like terms but have opposite signs, so they cancel each other out. Equally important is keeping track of your negative signs; subtraction changes the sign of each term in the subtracted polynomial. Properly combining like terms leads to a simplified and more digestible expression, enhancing comprehension and reducing complexity.

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This involves expanding brackets, combining like terms, and arranging terms for clarity. Simplification helps to present the expression in the most straightforward manner possible, aiding in the understanding and solving of algebraic problems.

The exercise provided results in \(4x - 4\) once simplified. To achieve simplicity, follow the steps carefully. First, expand the brackets as shown, ensuring all terms reflect the correct sign. Then, identify like terms and combine them. Finally, order the terms with the variable first and then the constant for a standard and neat final expression. It's a methodical way of streamlining an expression, making it easier to interpret or further manipulate if needed.