Questions: Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (y^2 - 9y + 14) / (y^2 + 5y - 14) Simplify the rational expression. Select the correct choice below and fill in any answer boxes in your choice. (y^2 - 9y + 14) / (y^2 + 5y - 14) = □ (Simplify your answer. Use positive exponents only.)

Solution Steps

To simplify the given rational expression, we need to factor both the numerator and the denominator. After factoring, we can cancel out any common factors. To find the numbers that must be excluded from the domain, we identify the values of \( y \) that make the denominator zero, as these would make the expression undefined.

To simplify the rational expression \(\frac{y^2 - 9y + 14}{y^2 + 5y - 14}\), we first factor both the numerator and the denominator.

• The numerator \(y^2 - 9y + 14\) factors to \((y - 7)(y - 2)\).
• The denominator \(y^2 + 5y - 14\) factors to \((y - 2)(y + 7)\).

After factoring, the expression becomes:

\[ \frac{(y - 7)(y - 2)}{(y - 2)(y + 7)} \]

We can cancel the common factor \((y - 2)\) from the numerator and the denominator:

\[ \frac{y - 7}{y + 7} \]

The values that make the denominator zero must be excluded from the domain. These are the solutions to the equation:

\[ y^2 + 5y - 14 = 0 \]

The factored form \((y - 2)(y + 7) = 0\) gives the solutions \(y = 2\) and \(y = -7\). Therefore, these values must be excluded from the domain.

Final Answer