Coefficient of x^2 in (((x – 2)^2 – 2)^2 – 2)^2

If the expression \( (((x – 2)^2 – 2)^2 – 2)^2 \) with three pairs of parentheses is multiplied out, what is the coefficient of \( x^2 \)?

Solution:

To solve this question, we can use the recursive process.

Let \(P_k(x)\) represent the polynomial after \(k\) squaring operations:
\( P_0(x) = x \)
\( P_{k+1}(x) = (P_k(x)\, – 2)^2 \quad \text{for } k = 0, 1, 2, 3, \dots \)

Let \(a_k\) be the coefficient of \(x^2\) in \(P_k(x)\). Then,

• Initial Step \((k=0)\):

\( P_0(x) = x \implies a_0 = 0 \)

• First Squaring \((k=1)\):

\( P_1(x) = (P_0(x)\, – 2)^2 \\ = (x\, – 2)^2 = x^2\, – 4x + 4 \\ \implies a_1 = 1 \\ \)

• Second Squaring \((k=2)\):

\( P_2(x) = (P_1(x)\, – 2)^2 \\ = (x^2 – 4x + 2)^2 \\ \)

The \(x^2\) terms come from:

• \( (-4x)^2 = 16x^2 \)
• \( 2 \times x^2 \times 2 = 4x^2 \)

Therefore, \(a_2 = 16 + 4 = 20 \)

• Third Squaring \((k=3)\):

\( P_3(x) = (P_2(x)\, – 2)^2 \\ = (x^4 – 8x^3 + 20x^2 – 16x + 2)^2 \\ \)

The \(x^2\) terms come from:

• \( (-16x)^2 = 256x^2 \)
• \( 2 \times 20x^2 \times 2 = 80x^2 \)

Therefore, \(a_3 = 256 + 80 = 336 \)

The coefficient of \(x^2\) in the fully expanded expression \( (((x – 2)^2 – 2)^2 – 2)^2 \) is \( \fbox{336} \).

Please let me know in the comments if you find any error in this solution.