Problem 19 - Precalculus Mathematics for Calculus

Question

Simplify the rational expression. $$\frac{y^{2}+y}{y^{2}-1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y}{y - 1}\).

Step 1: Factor the Numerator

Start by factoring the numerator \(y^2 + y\). You can do this by factoring out the common factor \(y\): \[ y^2 + y = y(y + 1) \]

Step 2: Factor the Denominator

Now, factor the denominator \(y^2 - 1\). Notice that \(y^2 - 1\) is a difference of squares, which can be factored as: \[ y^2 - 1 = (y - 1)(y + 1) \]

Step 3: Write the Expression with Factored Forms

Express the rational expression with the factored forms from Steps 1 and 2:\[ \frac{y(y + 1)}{(y - 1)(y + 1)} \]

Step 4: Cancel Common Factors

Identify and cancel out any common factors that appear in both the numerator and the denominator. In this case, \(y + 1\) appears in both, so you cancel it out:\[ \frac{y \cancel{(y + 1)}}{(y - 1) \cancel{(y + 1)}} = \frac{y}{y - 1} \]

Step 5: state the Simplified Expression

The simplified expression, after canceling out the common factor, is:\[ \frac{y}{y - 1} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra. It involves rewriting a polynomial as a product of its factors, which can help simplify complex expressions and solve equations more efficiently.
To factor a polynomial, you first look for common factors in each term. For example, in the expression \(y^2 + y\), the common factor is \(y\). By factoring out \(y\), you get \(y(y + 1)\). This process helps break down the polynomial into simpler components.
Factoring is not just about finding common terms; it can also involve recognizing special polynomial forms. Different types of factoring include:

• Finding a Greatest Common Factor (GCF)
• Using patterns, such as perfect square trinomials and the difference of squares (which we'll explore next)
• Factoring cubic and other higher-degree polynomials

Understanding how to factor effectively aids in simplifying rational expressions and solving equations, setting a strong foundation for more advanced mathematics.

Difference of Squares

The difference of squares is a specific type of polynomial that can be factored using a recognizable pattern.
A difference of squares is any expression of the form \(a^2 - b^2\). This can be factored into \((a + b)(a - b)\). For instance, in the exercise \(y^2 - 1\), both terms are perfect squares: \(y^2\) is \(y\) squared, and \(1\) is \(1\) squared.
Applying the difference of squares pattern, you factor \(y^2 - 1\) as \((y + 1)(y - 1)\).
Recognizing the difference of squares quickly allows you to break down expressions much faster and is particularly useful in simplifying rational expressions where these factors can often be canceled out with similar terms in numerators or other denominators.

Simplifying Expressions

Simplifying expressions is essential for making mathematical problems more approachable.
Once a rational expression has been factored, simplification involves canceling common factors in the numerator and denominator. This helps reduce the expression to its simplest form. In the original exercise, after factoring, the expression became \(\frac{y(y + 1)}{(y - 1)(y + 1)}\). The term \((y + 1)\) is common in both the numerator and the denominator, so it can be canceled, resulting in \(\frac{y}{y - 1}\).
However, it's important to remember that you can only cancel factors, not terms that are added or subtracted. Simplifying accurately requires care in distinguishing between these parts of an expression.

• Ensure all terms are fully factored.
• Cancel only factors that appear exactly the same in both the numerator and the denominator.
• Keep an eye on excluded values; for example, the expression is undefined if the denominator equals zero.

This skill is crucial for efficiently solving complex rational equations and algebraic operations.