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

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

Step 1: Rewrite the expression (x^2 - 4x + 4) in a simpler form. Notice that it can be factored as (x - 2)^2.
Step 2: Now, rewrite the original expression as ((x - 2)^2)^4. This simplifies to (x - 2)^8.
Step 3: Use the binomial expansion formula to expand (x - 2)^8. The general term in the expansion is given by T(k) = C(8, k) * (x)^k * (-2)^(8-k), where C(8, k) is the binomial coefficient.
Step 4: We want to find the term that is independent of x. This occurs when the power of x is 0, which means k must equal 8.
Step 5: Substitute k = 8 into the general term formula: T(8) = C(8, 8) * (x)^8 * (-2)^(8-8). This simplifies to T(8) = 1 * (x)^8 * (-2)^0 = 1 * 1 = 1.
Step 6: Now, we need to find the term independent of x. We need to find the term when k = 4, which gives us T(4) = C(8, 4) * (x)^4 * (-2)^(8-4).
Step 7: Calculate C(8, 4) = 70. Now substitute this into T(4): T(4) = 70 * (x)^4 * (-2)^4.
Step 8: Calculate (-2)^4 = 16. Therefore, T(4) = 70 * (x)^4 * 16.
Step 9: The term independent of x is when k = 4, which gives us 70 * 16 = 1120.

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.