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Factorization is the process of rewriting an expression as the product of its factors. This is an important skill in algebra, as it helps simplify complex expressions and solve equations more easily. In the example given, we started with the fraction \(\frac{x^2 + xy}{x^2}\). The numerator, \(x^2 + xy\), contains two terms. To factor the numerator, look for any common factors shared by these terms: \(x^2\) and \(xy\). Both terms have \(x\) as a common factor, which means we can factor \(x\) out of the expression. This gives us the factored form: \(x(x + y)\). By expressing the numerator in its simplest form, it becomes easier to work with when simplifying the fraction or solving further equations. Factorization is a crucial step for simplifying expressions and should never be skipped.

Simplifying algebraic expressions involves reducing them to their simplest form, making them easier to understand and work with. This often entails a combination of different techniques, such as factoring, distributing, and combining like terms. In our example, after factoring the numerator, we rewrite the fraction as \(\frac{x(x+y)}{x^2}\). To simplify, we identify any common terms in the numerator and the denominator. With \(x(x+y)\) and \(x^2\), we can spot that \(x\) is common to both. By canceling out these common terms, we reduce the complexity of the expression. This simplification step directly leads to a clearer and more concise expression: \(\frac{x+y}{x}\). Simplifying expressions is an essential skill in algebra, and it helps clarify the relationships between variables in an equation.

Canceling terms is an operation that simplifies algebraic fractions by eliminating common factors in the numerator and the denominator. The idea here is straightforward: if the same variable or factor is present in both the top and bottom of a fraction, they can "cancel" each other out. This reduces the expression to a simpler form. In our example, after factoring the numerator into \(x(x + y)\) and rewriting the fraction as \(\frac{x(x+y)}{x^2}\), we see that \(x\) is in both the numerator and denominator. By canceling \(x\) from both, we simplify the fraction to \(\frac{x+y}{x}\). This process should be done carefully, ensuring that only like terms are canceled. Canceling terms makes expressions easier to handle and is a common practice in solving algebraic equations.