91Ó°ÊÓ

Understanding how to combine like terms is essential when working with algebraic expressions, especially in operations with polynomials. Like terms in algebra are terms that contain the same variables to the same power. For instance, in the polynomial \(2x^3 + 3x^2 - 5x + x^3\), the terms \(2x^3\) and \(x^3\) are like terms because they both contain the variable \(x\) raised to the third power.

To combine like terms, simply add or subtract their coefficients. Following the above example, we combine \(2x^3\) and \(x^3\) to get \(3x^3\). This simplification makes the expression easier to work with and is a crucial step in polynomial operations such as addition, subtraction, and even more complex operations.

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. In the given exercise, subtraction is the focus. When subtracting polynomials, it is important to remember to distribute the negative sign to each term of the polynomial being subtracted before combining like terms.

For example, \(x^2 - 8x - 9\) \(-\) \(5x^2 - 4x - 3\) first requires us to change the subtraction into addition by distributing the negative sign, which flips the signs of the terms in the second polynomial. This leads to the new polynomial \(x^2 - 8x - 9 - 5x^2 + 4x + 3\). Now that we have a single polynomial, we can combine like terms. Not properly distributing the negative sign is a common error that can lead to an incorrect answer.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operators (like add, subtract, multiply, and divide). In the context of the exercise, the algebraic expressions are in the form of polynomials, which are expressions that include variables raised to whole number exponents and their coefficients.

The process of simplifying algebraic expressions often involves combining like terms and performing operations on polynomials, as demonstrated in the exercise. Polynomials can be as simple as \(2x + 3\) or as complex as \(4x^5 - x^3 + 3x^2 - 2x + 1\). The fundamental skills needed to manipulate these expressions are the same, and mastering them is key to success in algebra and beyond.