Questions: Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. y^2-y-56/y^2-13y+40 Simplify the rational expression. Select the correct choice below and fill in any answer boxes in your choice. y^2-y-56/y^2-13y+40 = (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. Once factored, we can cancel out any common factors. Additionally, we need to determine the values of \( y \) that make the denominator zero, as these values must be excluded from the domain.

We start with the rational expression:

\[ \frac{y^{2}-y-56}{y^{2}-13y+40} \]

We factor the numerator \(y^{2}-y-56\) and the denominator \(y^{2}-13y+40\):

• The numerator factors to \((y - 8)(y + 7)\).
• The denominator factors to \((y - 8)(y - 5)\).

Now we can rewrite the rational expression using the factored forms:

\[ \frac{(y - 8)(y + 7)}{(y - 8)(y - 5)} \]

We can cancel the common factor \((y - 8)\) (noting that \(y \neq 8\)):

\[ \frac{y + 7}{y - 5} \]

To find the values that must be excluded from the domain, we set the denominator equal to zero:

\[ y - 5 = 0 \quad \Rightarrow \quad y = 5 \]

Additionally, since we canceled \((y - 8)\), we also exclude \(y = 8\). Therefore, the excluded values are:

\[ y = 5, \quad y = 8 \]

Final Answer