Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Algebraic identities · 3 marks

James and Reshma were talking about algebraic identities they learnt in school.
James: (a − b)2 (a + b) = (a2 − 2ab + b2)(a + b)
Reshma: I have a different idea. (a − b)2 (a + b) = (a − b) [(a − b) (a + b)] = (a − b)(a2 − b2)
I will find this product to get the answer.
According to you, who is correct and why?
Try to combine more such identities and find new results.

Answer: Both are correct. Both ways give (a − b)2(a + b) = a3 − a2b − ab2 + b3. They just group the same three brackets differently; Reshma’s way is a little shorter (4 products instead of 6).

Step-by-step solution

Given: (a − b)2(a + b) = (a − b)(a − b)(a + b)
To find: Whose method is correct, and the expanded product

Idea: (a − b)2(a + b) is a product of three brackets. Multiplication can be grouped in any order (associative property), so James (first squaring (a − b)) and Reshma (first using (a − b)(a + b) = a2 − b2) must reach the same answer.

  1. James: (a2 − 2ab + b2)(a + b)
    = a3 + a2b − 2a2b − 2ab2 + ab2 + b3
    = a3 − a2b − ab2 + b31 mark
  2. Reshma: (a − b)(a2 − b2) = a3 − ab2 − a2b + b3
    = a3 − a2b − ab2 + b31 mark
  3. Both answers are the same, so both are correct. The reason: multiplying three brackets in a different grouping does not change the product. Reshma’s route uses a known identity first, so there are fewer terms to multiply.½ mark
  4. Combining more identities (one new result): (a + b)2(a − b) = (a + b)(a2 − b2) = a3 + a2b − ab2 − b3. Another: (a + b)2(a − b)2 = [(a + b)(a − b)]2 = (a2 − b2)2 = a4 − 2a2b2 + b4.½ mark
Both James and Reshma are correct: (a − b)²(a + b) = a³ − a²b − ab² + b³ either way. New results: (a + b)²(a − b) = a³ + a²b − ab² − b³ and (a + b)²(a − b)² = a⁴ − 2a²b² + b⁴.

Check: a = 3, b = 1: (2)2(4) = 16, and 27 − 9 − 3 + 1 = 16 ✓.

Answer to write in the exam

James: (a2 − 2ab + b2)(a + b) = a3 + a2b − 2a2b − 2ab2 + ab2 + b3 = a3 − a2b − ab2 + b3

Reshma: (a − b)(a2 − b2) = a3 − ab2 − a2b + b3 = a3 − a2b − ab2 + b3

Same result (grouping the factors differently does not change a product)

New results: (a + b)2(a − b) = (a + b)(a2 − b2) = a3 + a2b − ab2 − b3

(a + b)2(a − b)2 = (a2 − b2)2 = a4 − 2a2b2 + b4

∴ Both James and Reshma are correct: (a − b)2(a + b) = a3 − a2b − ab2 + b3

Common mistakes that cost marks

  • Thinking only one method can be right. Different groupings of a product always give the same answer.
  • In James’s method, combining a2b − 2a2b as +a2b. It is −a2b.
  • Writing (a − b)(a + b) = a2 + b2. It is a2 − b2.

How this can come in the exam

MCQ (1 mark)

(a − b)2(a + b) is equal to

  1. a3 − b3
  2. a3 − a2b − ab2 + b3
  3. a3 + a2b − ab2 − b3
  4. a3 − 3a2b + 3ab2 − b3
Show answer

(B) a3 − a2b − ab2 + b3
(a − b)(a2 − b2) = a3 − ab2 − a2b + b3.

Short answer (2 marks)

Using a suitable identity, find (x + 2)2(x − 2)2.

Show answer(x + 2)2(x − 2)2 = [(x + 2)(x − 2)]2 = (x2 − 4)2 (1 mark) = x4 − 8x2 + 16 (1 mark).

Try one yourself

Find (x + 3)2(x − 3) in two different ways and check that they agree.

Show answer

(x + 3)(x2 − 9) = x3 − 9x + 3x2 − 27; and (x2 + 6x + 9)(x − 3) = x3 + 3x2 − 9x − 27. Both give x3 + 3x2 − 9x − 27.

More questions like this

All Algebraic identities questions · All maths questions

← Factor the following:What do you think (a + b)³ will look like? →