If a + b = 5 and ab = 6, Find: (a³ + b³)² − 9a²b² (a + b)².

Correct — A. Find a³ + b³ first, using a³ + b³ = (a + b)³ − 3ab(a + b).

a³ + b³ = 5³ − 3 × 6 × 5 = 125 − 90 = 35
First term: (a³ + b³)² = 35² = 1225
Second term: 9a²b²(a + b)² = 9 × 36 × 25 = 8100

Subtract: 1225 − 8100 = −6875 → option (a)

Why the others are wrong

• (b)
  6875 — 6875 has the right size but the wrong sign. The second term, 8100, is larger than the first, 1225, so the difference must be negative.
• (c)
  −3876 — −3876 would need the second term to be 1225 + 3876 = 5101. But 9 × 6² × 5² = 8100, so the difference is −6875.
• (d)
  3876 — 3876 is positive, and 8100 > 1225 forces a negative answer. Its size is also off: the gap between the two terms is 6875.

When a + b and ab are given, any symmetric expression in a and b can be built from them. Expanding (a + b)³ = a³ + b³ + 3ab(a + b) gives a³ + b³ = (a + b)³ − 3ab(a + b).

The expression is also a difference of squares: [a³ + b³]² − [3ab(a + b)]², because 9a²b²(a + b)² = [3ab(a + b)]².

So it factors as (35 − 90)(35 + 90) = −55 × 125 = −6875, the same result.

Key facts

• a³ + b³ = (a + b)³ − 3ab(a + b).
• a² + b² = (a + b)² − 2ab.
• x² − y² = (x − y)(x + y).

Common traps

• Dropping the (a + b) factor: (a + b)³ − 3ab = 125 − 18 = 107, not 35.
• Squaring only part of the second term. In 9a²b²(a + b)², ab and a + b are both squared: 9 × 36 × 25.

No directly related past PYQ was found.