Final answer:
The coefficient of x⁶ in the expansion of (x² - 3)⁸ is -13,608, which is found using the binomial theorem and substituting the appropriate values for k.
Step-by-step explanation:
To find the coefficient of x^n in the expansion of (x² - 3)⁸, we can use the binomial theorem. Since n = 6, we are looking for the term that contains x to the sixth power. In the binomial expansion of (x² - 3)⁸, the general term is given as:
T(k+1) = C(8, k) * (x²)^(8-k) * (-3)^k
Now we need to find the value of k such that the exponent of x is 6. Because the x term in our binomial is x², we are effectively looking for the power of x² that results in x⁶. This means:
(x²)^(8-k) = x⁶ => 2(8-k) = 6 => 16 - 2k = 6 => 2k = 10 => k = 5
Substituting k = 5 into the general term to find the coefficient:
T(5+1) = C(8, 5) * (x²)^(8-5) * (-3)^5 = C(8, 5) * x⁶ * (-243) = 56 * x⁶ * -243
So, the coefficient of x⁶ in the expansion of (x² - 3)⁸ is -13,608.