Exercise 3: Eight Recursive Calls on Half-Sized Subproblems — Possible Solution
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GIVEN
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8 recursive calls, each on a half-sized (n/2) subproblem, plus O(n^2)
non-recursive work.

STEP 1: WRITING THE RECURRENCE
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Following this chapter's own general shape, and Chapter 5's own
translation method:

  T(n) = 8T(n/2) + n^2

STEP 2: IDENTIFYING a, b, AND f(n)
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a = 8
b = 2
f(n) = n^2

STEP 3: COMPUTING n^(log_b a)
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log_b(a) = log base 2 of 8 = 3

n^(log_b a) = n^3

STEP 4: COMPARING f(n) TO n^(log_b a)
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f(n) = n^2
n^(log_b a) = n^3

n^3 grows faster than n^2 - the recursive branching term wins here.

STEP 5: IDENTIFYING THE CASE AND THE RESULT
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Since n^(log_b a) grows faster than f(n), per this chapter's own
Master Theorem table this is CASE 1 - the recursion dominates the
non-recursive work entirely.

  T(n) = Theta(n^(log_b a)) = Theta(n^3)

WHY THIS WORKS AS AN ANSWER
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The recurrence is constructed directly from the given description
using Chapter 5's own translation method, a, b, and f(n) are
identified explicitly, n^(log_b a) is computed precisely (log base 2
of 8 equals exactly 3, not estimated), and the case is selected by
directly comparing the two growth rates per this chapter's own three-
case table.