Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $\frac{y^2 + 7y - 18}{y^2 - 3y + 2}$

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational expression. For the numerator \(y^2 + 7y - 18\), find two numbers that multiply to \(-18\) and add to \$7\(. For the denominator \)y^2 - 3y + 2\(, find two numbers that multiply to \)2\( and add to \)-3$.
2. Rewrite the numerator and denominator as products of their factors. For example, the numerator will be factored into the form \((y + a)(y + b)\) and the denominator into \((y + c)(y + d)\), where \(a\), \(b\), \(c\), and \(d\) are the numbers found in the previous step.
3. Simplify the rational expression by canceling out any common factors that appear in both the numerator and the denominator.
4. Identify the values of \(y\) that make the original denominator equal to zero, since these values are excluded from the domain. Set the original denominator equal to zero and solve for \(y\).
5. State the simplified rational expression along with the domain restrictions, which exclude the values of \(y\) found in the previous step.

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions involves factoring the numerator and denominator and then canceling out common factors. This process reduces the expression to its simplest form, making it easier to analyze or use in further calculations.

Factoring Quadratic Expressions

Factoring quadratics means rewriting a quadratic polynomial as a product of two binomials. Recognizing patterns like factoring trinomials into (y + m)(y + n) where m and n multiply to the constant term and add to the linear coefficient is essential for simplifying rational expressions.

Domain Restrictions in Rational Expressions

The domain of a rational expression excludes values that make the denominator zero, as division by zero is undefined. Identifying these excluded values requires setting the denominator equal to zero and solving for the variable to find restrictions on the domain.