Final answer:
To find the coefficient of x²⁸, we combine terms from the two polynomials that will multiply together to get x²⁸. Systematically applying this method, we find that the coefficient is 224. The correct answer is B. 224.
Step-by-step explanation:
To find the coefficient of x²⁸ in the expansion of (1+x+x²+....+x²⁷)(1+x+x²+....+x¹⁴)², we can use the binomial theorem. The binomial theorem states that the coefficient of a term in the expansion of (a + b)ⁿ can be found using the formula C(n, k) * a^(n-k) * b^k, where C(n, k) is the number of ways to choose k items from a set of n items. In this case, we have two sets of terms: (1+x+x²+....+x²⁷) and (1+x+x²+....+x¹⁴), which we can consider as the terms a and b respectively.
Using the formula, the coefficient of x²⁸ will be C(28, k) * a^(28-k) * b^k, where k is the number of terms from the set (1+x+x²+....+x¹⁴) in the expansion of (1+x+x²+....+x²⁷)(1+x+x²+....+x¹⁴)² that we need to choose.
To find the coefficient of x²⁰ in the expansion of (1+x+x²+....+x₇)(1+x+x²+....+x⁺)², we need to consider how we can form x²⁰ by multiplying terms in the two polynomials.
Firstly, notice that (1+x+x²+....+x⁺)² equals the sum of 1+2x+3x²+...+15x⁺. Then consider the first polynomial (1+x+x²+....+x₇); we can pick out the appropriate power of x that when multiplied by a term in the second polynomial gives us x²⁰. For example, x⁻ from the first times x₇ from the square of the second equals x²⁰.
By systematically calculating the coefficients for each product term that could result in an x²⁰ term, we find that the coefficient is 224. Option B is the correct answer.