Find the term independent of x in the expansion of (x^2 - 4x + 4)^6. (2020)

Find the term independent of x in the expansion of (x^2 - 4x + 4)^6. (2020)

Step 1: Rewrite the expression (x^2 - 4x + 4) as (x - 2)^2. This is because (x - 2)(x - 2) = x^2 - 4x + 4.
Step 2: Now, we need to expand (x - 2)^6 using the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = x and b = -2, and n = 6. So we will expand (x - 2)^6.
Step 4: We are looking for the term that does not contain x, which is called the term independent of x. This happens when the power of x is zero.
Step 5: The general term in the expansion is given by T(k) = 6Ck * (x)^(6-k) * (-2)^k.
Step 6: To find the term independent of x, set the exponent of x to zero: 6 - k = 0, which means k = 6.
Step 7: However, we need to find the term independent of x, so we actually need to find when the power of x is zero. This occurs when k = 3 (since 6 - 3 = 3).
Step 8: Calculate the coefficient for k = 3: T(3) = 6C3 * (x)^(6-3) * (-2)^3.
Step 9: Calculate 6C3, which is 20. Then calculate (-2)^3, which is -8.
Step 10: Multiply these together: 20 * (-8) = -160.
Step 11: The term independent of x is the constant term, which is 24.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding the Constant Term – Identifying the term in a polynomial expansion that does not contain the variable, in this case, x.