91Ó°ÊÓ

When working with algebra, we often encounter algebraic expressions, which are a combination of numbers, variables (such as \( y \)), and arithmetic operations like addition, subtraction, multiplication, and division. In the given exercise, \( \frac{y^{3}+y}{y^{2}-y} \) is an example of an algebraic expression. Understanding these expressions is crucial for simplifying and manipulating them to solve various algebraic problems.

Algebraic expressions can take many forms, such as polynomials, which are expressions including terms with variables raised to whole number exponents and having coefficients. In the exercise, each term within the division problem is part of a polynomial expression. Being able to recognize and manipulate these expressions is a foundational skill in algebra.

The process of factoring polynomials is similar to breaking down a number into its prime factors. When we factor a polynomial, we are looking for simpler polynomial expressions that, when multiplied together, give us the original polynomial. This skill is essential when simplifying complex algebraic expressions and solving equations.

In our example, we factorized \(y^{3} + y\) into \(y(y^{2} + 1)\) and \(y^{3} - y^{2}\) into \(y^{2}(y - 1)\). By doing so, we transformed these polynomials into a product of their factors, which can greatly simplify the expression, especially when dividing polynomials. To factor effectively, one must look for common factors, use identities, or apply special patterns such as the difference of squares.

Simplifying algebraic fractions involves reducing the complexity of a fraction while maintaining its value. This can include reducing common factors between the numerator and the denominator, much like simplifying numerical fractions. The goal is to achieve the simplest form possible.

After factoring the polynomials in our exercise, we saw terms cancel out because they appeared in both the numerator and the denominator. Through simplification, the expression \( \frac{y(y^{2}+1)}{y(y-1)} \times \frac{(y-1)^{2}}{y^{2}(y-1)} \) was reduced to \( \frac{(y^{2}+1)(y-1)}{y} \). This step is often the final push in transforming a daunting fraction into an accessible equation ready for further analysis or plotting.