To find the least common denominator (LCD) for the fractions
\[ \frac{3}{5y + 40} \quad \text{and} \quad \frac{7}{y^2 + 11y + 24}, \]
we need to factor the denominators of both fractions.
- Factoring \( 5y + 40 \):
The expression \( 5y + 40 \) can be factored as:
\[ 5(y + 8). \]
- Factoring \( y^2 + 11y + 24 \):
Next, we need to factor the quadratic \( y^2 + 11y + 24 \). We are looking for two numbers that multiply to \( 24 \) (the constant term) and add to \( 11 \) (the coefficient of \( y \)). The numbers \( 3 \) and \( 8 \) work, so we can factor it as:
\[ (y + 3)(y + 8). \]
- Finding the LCD:
Now that we have both factored forms:
- \( 5(y + 8) \) for the first fraction
- \( (y + 3)(y + 8) \) for the second fraction
The LCD needs to include all unique factors:
- From \( 5(y + 8) \), we have \( 5 \) and \( (y + 8) \).
- From \( (y + 3)(y + 8) \), we have \( (y + 3) \) and \( (y + 8) \).
To find the LCD, we take the highest power of each unique factor:
\[ \text{LCD} = 5(y + 8)(y + 3). \]
Thus, the least common denominator (LCD) for the fractions is:
\[ 5(y + 8)(y + 3). \]