Question

Find the LCD for the fractions.
StartFraction 3 Over 5 y plus 40 EndFraction
​, StartFraction 7 Over y squared plus 11 y plus 24 EndFraction
  

Answer 1

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.

1. **Factoring \( 5y + 40 \)**:

The expression \( 5y + 40 \) can be factored as:

\[
5(y + 8).
\]

2. **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:

\[
(y + 3)(y + 8).
\]

3. **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:

\[
5(y + 8)(y + 3).
\]