Problem 20 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{x y+y^{2}}{x^{2}-y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y}{x-y} \).

Step 1: Identifying the Expression

The given rational expression is \( \frac{x y + y^2}{x^2 - y^2} \). Our goal is to simplify this expression by factoring both the numerator and the denominator.

Step 2: Factoring the Numerator

Observe the numerator \( xy + y^2 \). Notice that it can be factored by taking \( y \) common, giving us \( y(x + y) \).

Step 3: Factoring the Denominator

The denominator \( x^2 - y^2 \) is a difference of squares. This can be factored into \( (x-y)(x+y) \).

Step 4: Writing the Expression with Factors

Replace both, the numerator and the denominator with their factored forms. The expression becomes: \( \frac{y(x+y)}{(x-y)(x+y)} \).

Step 5: Canceling Common Factors

Observe that \( (x+y) \) is a common factor in both the numerator and the denominator. We can cancel out this common factor, simplifying the expression to \( \frac{y}{x-y} \).

Step 6: State the Simplified Expression

After canceling the common terms, the simplified form of the given rational expression is \( \frac{y}{x-y} \).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in simplifying rational expressions. It involves breaking down a polynomial into a product of simpler polynomials. For example, consider the polynomial found in the numerator of our expression:

• Given: \(xy + y^2\)
• Look for a common factor in all the terms. Here, both terms have a common factor of \(y\).

By factoring out \(y\), the expression becomes \(y(x+y)\). This process transforms the original polynomial into a simpler form that helps in further simplification. Becoming comfortable with recognizing common factors and applying factorization is key. Once a polynomial is factored, it becomes significantly easier to simplify the entire rational expression.

Difference of Squares

The difference of squares is a special case in polynomial factorization, where a polynomial takes the form of \(a^2 - b^2\). This can be factored into \((a-b)(a+b)\). Let's look at the denominator in our expression:

• Given: \(x^2 - y^2\)
• Recognize it as a difference of squares since \(x^2\) and \(y^2\) are both perfect squares.

Applying the formula, this becomes \((x-y)(x+y)\). Understanding this pattern is immensely useful, as it allows for quick and straightforward factoring. Recognizing when to use this rule helps in simplifying expressions quickly and efficiently, saving time and effort in calculations.

Canceling Common Factors

Canceling common factors is the final step after factoring both the numerator and the denominator. This step involves simplifying the expression by removing duplicate factors that appear in both parts. For our expression, after factoring, we have:

• Numerator: \(y(x+y)\)
• Denominator: \((x-y)(x+y)\)
• Common Factor: \((x+y)\)

By canceling the common factor \((x+y)\), the expression is simplified to \(\frac{y}{x-y}\). It's important to only cancel common factors that appear in both the numerator and denominator. This simplification reduces the complexity of the rational expression, making it easier to understand and work with. Always verify that cancellation is possible and valid to maintain the mathematical integrity of the expression.