Final answer:
The coefficient of x² - y⁹ + x² -y⁹ in the expansion of (2x−2y)¹¹ is zero because such a term does not exist in the binomial expansion where the exponents must sum to 11.
Step-by-step explanation:
Coefficient
of x² - y⁹ + x² -y⁹ in the expansion of (2x−2y)¹¹:
To find the coefficient of a particular term in the expansion of a binomial expression raised to a power, we use the binomial theorem. However, in this case, we can observe that the term x² does not appear in the expansion of (2x - 2y)¹¹ directly because the exponents in the binomial expansion must add up to 11. Therefore, any term containing x² will also have y¹ or a lower power of y due to the binomial expansion.
The given expression 'x² - y⁹ + x² -y⁹' simplifies to '2x² - 2y⁹', and there is no such term in the expansion of (2x−2y)¹¹. Hence, the coefficient of x² - y⁹ + x² -y⁹ in the expansion is simply zero.