Final answer:
Using the binomial theorem to expand (a + x)⁵, we find the constant term to be a⁵. Multiplying this by the constant term 1 from 1 + (2x/a), the result is a nonzero constant, a⁵, as long as a is nonzero. Therefore, the correct answer is option a: the expansion has a nonzero constant.
Step-by-step explanation:
The question is asking whether the expression [(1 + 2x/a)(a + x)⁵] has a nonzero constant term after expansion. To tackle this problem, we use the binomial theorem to expand the term (a + x)⁵. According to the binomial theorem, (a + b)⁵ will give us a series of terms that involve powers of a and b. In our case, since we are looking for the constant term, we are interested in the term where x is raised to the power of 0, which occurs when the entire term is a⁵. Now, since there's a multiplier of 1 + (2x/a) outside the binomial expansion, to find if there's a constant term, we must also consider this multiplier.
The constant term in (a + x)⁵ is a⁵. Multiplying this by the constant term in 1 + (2x/a), which is simply 1, we get a constant term of a⁵*1, which is a⁵. As long as a is nonzero, then a⁵ will also be nonzero. Therefore, the expansion has a nonzero constant, which answers the student's question with option a.