To determine the absolute value of the difference between the coefficients of
x4 and
x6 in the expansion of the given expression, we start with:
x2−2x2+(x+1)4(x2−1)2We simplify the function as follows:
f(x)=‌Expanding, we have:
f(x)=2x2(x2+1)−1(x2+2)−1Simplifying further gives:
f(x)=‌(1+x2)−1(1+‌)−1This can be expanded into:
f(x)=x2[1−x2+x4−x6+⋯]⋅[1−‌+‌−‌+⋯]Now, let's find the coefficients.
For
x4 :
Coefficient calculation:
[−‌−1]=‌For
x6 :
Coefficient calculation:
[‌+‌+1]−‌Now calculate the absolute difference:
‌‌ Difference ‌=|‌−(‌)|=|‌+‌|‌=|‌+‌|=|‌|=‌This process allows us to correctly determine the absolute value of the difference between the coefficients of
x4 and
x6 in the expansion.