Minus b

The idea

Replacing b by –b in the identity for (a + b)³ gives (a – b)³ = a³ – 3a²b + 3ab² – b³, whose terms are alternately positive and negative.

Write −b for b

We have (a + b)³ = a³ + 3a²b + 3ab² + b³. What happens if we write −b in place of b?

How many of the four terms of (a − b)³ will be negative?

• 1
• 2
• 4

Choose the signs

Each card shows one term after the swap. Choose its sign. Think: an odd power of −b is negative and an even power is positive.

Alternating signs

(a − b)³ = a³ − 3a²b + 3ab² − b³. Of the four terms, two are positive and two are negative, and they appear alternately.

(a + (−b))³ = a³ + 3a²(−b) + 3a(−b)² + (−b)³ Because (−b)² = b² and (−b)³ = −b³, we get a³ − 3a²b + 3ab² − b³.

Notes

(a − b)³ = a³ − 3a²b + 3ab² − b³: write −b for b in (a + b)³. The signs alternate.

Check yourself

Which is the expansion of (x − 1)³?

For a = 5 and b = 2, 125 − 150 + 60 − 8 is (a − b)³ written out. What is its value?

Answer: 27

125 − 150 + 60 − 8 = 27, which is also (5 − 2)³ = 3³.

In a³ − 3a²b + 3ab² − b³, which terms are negative?

8n³ − 60n²m + 150nm² − 125m³ is of the form (a − b)³. What are a and b?

• x³ − 3x² − 3x − 1. The signs alternate. The third term is +3x because (−1)² = +1.
• x³ − 1. The two middle terms −3x² and +3x are missing.
• x³ − 3x² + 3x − 1 — correct. Yes: a = x and b = 1, with signs +, −, +, −.
• The 2nd and the 4th — correct. Yes. The signs alternate: +, −, +, −.
• The 2nd, 3rd and 4th. The third term is +3ab², because (−b)² = +b².
• Only the 4th. The second term 3a²(−b) = −3a²b is negative too.
• a = 8n and b = 125m. These are the cubes: a³ = 8n³ gives a = 2n, and b³ = 125m³ gives b = 5m.
• a = 2n and b = 5m — correct. Yes: (2n)³ − 3(2n)²(5m) + 3(2n)⁠(5m)² − (5m)³ = (2n − 5m)³.
• a = 2n and b = 25m. b³ = 125m³, so b = 5m. (25 is 5².)