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
Transcript text: 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
Solution Steps
Step 1: Factorization of Polynomials
The given expressions are 7a+21 and a2+3a. We factor each polynomial:
For 7a+21:
7a+21=7(a+3)
For a2+3a:
a2+3a=a(a+3)
Step 2: Identify Common Factors
From the factorization, we have:
7(a+3)
a(a+3)
The common factor in both expressions is (a+3).
Step 3: Determine the Least Common Denominator
To find the least common denominator (LCD), we take the product of the highest powers of all factors present in the factorizations:
LCD=7⋅a⋅(a+3)
Final Answer
Thus, the least common denominator of the given fractions is:
7a(a+3)