Question

Simplify the rational expression.

$\frac{y^2+y}{y^2-1}$

Short answer

The simplified rational expression is$\frac{y}{y-1}$

Step-by-step solution

Step 1. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Step 2. Analyze the expression.

Given$\frac{y^2+y}{y^2-1}$

This is a rational expression.

To simplify the rational expression, we factor both numerator and denominator and use this$\frac{AC}{BC}=\frac{A}{B}$

Therefore, to simplify a rational expression, we cancel factors that are common to the numerator and denominator.

Step 3. Simplify the expression.

Consider

$\frac{y^2+y}{y^2-1}=\frac{y\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}\mathrm{..................}\left[a^2-b^2=\left(a-b\right)\left(a+b\right)\right]$

The common factor to the numerator and denominator is$y+1$

So we will cancel the common factor from the numerator and denominator.

The simplified expression is$\frac{y\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}=\frac{y}{y-1}$