Problem 12 - Algebra and Trigonometry

Question

simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{y^{2}-4 y-5}{y^{2}+5 y+4} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{y-5}{y+4} \), and the values that should be excluded from the domain are y = -4 and y = -1.

Step 1: Factorization

The first step is to factorize both the numerator and the denominator. After factorization, the expression becomes: \( \frac{(y-5)(y+1)}{(y+4)(y+1)} \)

Step 2: Simplification

Simplify the expression by cancelling out common factors in the numerator and denominator. Here, (y + 1) is the common factor that can be cancelled. The simplified expression thus becomes: \( \frac{y-5}{y+4} \)

Step 3: Identify Exclusions

From the simplified expression, y cannot be -4 because this makes the denominator zero. Additionally, from the original expression before simplification, y cannot be -1, as it is a factor cancelled from the denominator, making the denominator zero.

Key Concepts

Factorization

Factoring is an essential skill when working with rational expressions. It involves breaking down algebraic expressions into products of simpler expressions. In our exercise example, the numerator \(y^2-4y-5\) and the denominator \(y^2+5y+4\) are both quadratic expressions that can be factored into binomials.

For the numerator, we're looking for two numbers that multiply to -5 and add to -4. These numbers are -5 and +1, so the numerator can be factored into \((y-5)(y+1)\). In a similar fashion, we're seeking two numbers that multiply to +4 and add to +5 for the denominator, which are +4 and +1. Thus, the denominator is factored into \((y+4)(y+1)\).

Understanding factorization is crucial because it allows us to simplify rational expressions by canceling out common factors between the numerator and denominator, just as \((y+1)\) was canceled in our example. This simplification can help make complex problems more manageable.

Excluded Values

When working with rational expressions, it's important to identify excluded values, also known as the restrictions for the variable. These are values that would make the denominator equal to zero, which is undefined in mathematics. An undefined denominator results in an expression with no meaningful value, often referred to as a singularity or discontinuity.

For the simplified expression \(\frac{y-5}{y+4}\), the excluded value is -4 because substituting -4 for y in the denominator would result in 0, making the expression undefined. However, we must also consider the original expression before simplification. Before we canceled the common factor \((y+1)\), y could not equal -1 since that would have also resulted in zero in the denominator.

Identifying excluded values helps avoid undefined results and ensures the domain of the expression is accurately defined, which is vital in various applications such as calculus and algebraic graphing.

Rational Expressions Domain

The domain of a rational expression refers to the set of allowable values that the variable can have. In other words, it includes all real numbers except those that make the denominator zero. Establishing the domain is essential not just for correct computation, but also for understanding the behavior of the expression across different values of the variable.

In the context of our exercise, after determining the excluded values of -4 and -1, we can define the domain of the simplified expression \(\frac{y-5}{y+4}\) as 'all real numbers except y is not equal to -4.' When working with rational expressions, always remember to consider the original, non-simplified form to ensure no excluded values are overlooked. This careful treatment allows for more accurate graphing and function analysis in further studies of mathematics.