Question

In the expansion of (3a + 4b)8, which of the following are possible variable terms? Explain your reasoning.

A) a²b³
B) a⁵b³
C) ab⁸
D) b⁸
E) a⁴b⁴
F) a⁸
G) ab⁷
H) a⁶b⁵

Answer 1

Possible variable terms in the expansion of (3a + 4b)8 are a2b3, a4b4, and a6b5.

Step-by-step explanation:

In the expansion of (3a + 4b)8, possible variable terms can be determined using the binomial theorem. According to the theorem, each term in the expansion will have the form (3a)p(4b)q, where p and q are non-negative integers and p + q = 8. The variables a and b will be raised to powers p and q, respectively. Using this information, we can analyze each option:

1. A) a2b3 - This is a possible term because p = 2 and q = 3 satisfy p + q = 5.
2. B) a5b3 - This is not a possible term because p = 5 and q = 3 do not satisfy p + q = 8.
3. C) ab8 - This is not a possible term because p = 1 and q = 8 do not satisfy p + q = 8.
4. D) b8 - This is not a possible term because p = 0 and q = 8 do not satisfy p + q = 8.
5. E) a4b4 - This is a possible term because p = 4 and q = 4 satisfy p + q = 8.
6. F) a8 - This is not a possible term because p = 8 and q = 0 do not satisfy p + q = 8.
7. G) ab7 - This is not a possible term because p = 1 and q = 7 do not satisfy p + q = 8.
8. H) a6b5 - This is a possible term because p = 6 and q = 2 satisfy p + q = 8.

Therefore, the possible variable terms in the expansion of (3a + 4b)

8

are A) a

2

b

3

, E) a

4

b

4

, and H) a

6

b

5

.