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Algebraic identities · 2 marks

What do you think (a + b)3 will look like?

Answer: (a + b)3 = a3 + 3a2b + 3ab2 + b3

Step-by-step solution

Given: (a + b)2 = a2 + 2ab + b2
To find: The expansion of (a + b)3

Idea: A cube is a square times one more factor: (a + b)3 = (a + b)(a + b)2. Use the square identity, then the distributive property.

  1. (a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2)½ mark
  2. Multiply by a: a3 + 2a2b + ab2
    Multiply by b: a2b + 2ab2 + b3½ mark
  3. Add like terms: a2b-terms 2 + 1 = 3; ab2-terms 1 + 2 = 3.½ mark
  4. (a + b)3 = a3 + 3a2b + 3ab2 + b3. This is a new identity.½ mark
(a + b)³ = a³ + 3a²b + 3ab² + b³.

Check: a = 2, b = 1: 33 = 27 and 8 + 12 + 6 + 1 = 27 ✓.

Answer to write in the exam

(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2)

= a3 + 2a2b + ab2 + a2b + 2ab2 + b3

∴ (a + b)3 = a3 + 3a2b + 3ab2 + b3

Common mistakes that cost marks

  • Writing (a + b)3 = a3 + b3. The two middle terms 3a2b + 3ab2 are missing.
  • Writing the middle coefficients as 2 (copying from the square). For the cube they are 3 and 3.
  • Writing 3a2b2: the terms are 3a2b and 3ab2; in each term the powers add up to 3.

How this can come in the exam

MCQ (1 mark)

(x + 2)3 equals

  1. x3 + 8
  2. x3 + 6x2 + 12x + 8
  3. x3 + 3x2 + 3x + 8
  4. x3 + 6x2 + 6x + 8
Show answer

(B) x3 + 6x2 + 12x + 8
x3 + 3x2(2) + 3x(4) + 8 = x3 + 6x2 + 12x + 8.

Short answer (2 marks)

Find 113 using (a + b)3 = a3 + 3a2b + 3ab2 + b3.

Show answer(10 + 1)3 = 1000 + 3(100)(1) + 3(10)(1) + 1 (1 mark) = 1000 + 300 + 30 + 1 = 1331 (1 mark).

Try one yourself

Expand (2x + 1)3.

Show answer

(2x)3 + 3(2x)2(1) + 3(2x)(1)2 + 1 = 8x3 + 12x2 + 6x + 1.

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