Questions: Assume that the expressions given are denominators of fractions. Find the least common denominator. 7 a+21, a^2+3 a A. 7 a^2+3 B. 7 a^2+21 C. 7 a(a+3) D. 7 a+3

Solution Steps

The given expressions are $7a + 21$ and $a^2 + 3a$. We factor each polynomial:

1. For $7a + 21$: $$7a + 21 = 7(a + 3)$$

2. For $a^2 + 3a$: $$a^2 + 3a = a(a + 3)$$

From the factorization, we have:

• $7(a + 3)$
• $a(a + 3)$

The common factor in both expressions is $(a + 3)$.

To find the least common denominator (LCD), we take the product of the highest powers of all factors present in the factorizations:

$$\text{LCD} = 7 \cdot a \cdot (a + 3)$$

Final Answer