Answer:
The middle term is [tex]-160x^{3}[/tex]
Step-by-step explanation:
Given
[tex](\frac{2}{x} - x^2)^6[/tex]
Required
Determine the middle term
This can be expanded using binomial expansion
i.e.
[tex](a + b)^n = a^n + ^nC_1a^{n-1}b^1 + ^nC_2a^{n-2}b^2 +....+b^n[/tex]
When n = 6;
[tex](a + b)^6 = a^6 + ^6C_1a^{6-1}b^1 + ^6C_2a^{6-2}b^2 + ^6C_3a^{6-3}b^3+ ^6C_4a^{6-4}b^4+ ^6C_5a^{6-5}b^5+b^n[/tex]
[tex](a + b)^6 = a^6 + ^6C_1a^{5}b^1 + ^6C_2a^{4}b^2 + ^6C_3a^{3}b^3+ ^6C_4a^{2}b^4+ ^6C_5a^{1}b^5+b^6[/tex]
[tex](a + b)^6 = a^6 + ^6C_1a^{5}b + ^6C_2a^{4}b^2 + ^6C_3a^{3}b^3+ ^6C_4a^{2}b^4+ ^6C_5ab^5+b^6[/tex]
The middle term here is given as:
[tex](a + b)^6 = ^6C_3a^{3}b^3[/tex]
In [tex](\frac{2}{x} - x^2)^6[/tex]
[tex]a = \frac{2}{x}[/tex] and [tex]b = -x^2[/tex]
This gives:
[tex](\frac{2}{x} - x^2)^6 = ^6C_3(\frac{2}{x})^{3}(-x^2)^3[/tex]
[tex](\frac{2}{x} - x^2)^6 = 20*(\frac{2}{x})^{3}(-x^2)^3[/tex]
Solve all exponents
[tex](\frac{2}{x} - x^2)^6 = 20*\frac{8}{x^3}*(-x^6)[/tex]
[tex](\frac{2}{x} - x^2)^6 = -\frac{20*8*x^6}{x^3}[/tex]
[tex](\frac{2}{x} - x^2)^6 = -160*x^{6-3}[/tex]
[tex](\frac{2}{x} - x^2)^6 = -160*x^{3}[/tex]
[tex](\frac{2}{x} - x^2)^6 = -160x^{3}[/tex]
Hence, the middle term is [tex]-160x^{3}[/tex]
