Problem: The coefficient of x7 in the expansion of (2x2−x2)8 is:
Answer Choices:
A. 56
B. −56
C. 14
D. −14
E. 0
Solution:
(2x2−x2)8=(2x)81(x3−4)8=(2x)81[(x3)8+8(x3)7(−4)+1⋅28⋅7(x3)6(−4)2+1⋅2⋅38⋅7⋅6(x3)5(−4)3+…]
The required term is 28⋅x81⋅1⋅2⋅38⋅7⋅6(x3)5(−4)3; the required coefficient is −14
or
28x81C(8,5)(x3)5(−4)3=28x81⋅1⋅2⋅38⋅7⋅6(−4)3x7=−14x7; the required coefficient is −14.