Question

Factorise each of the following: 

(i) 8a ³ + b ³ + 12a 2b + 6ab² 

(ii) 8a ³ – b ³ – 12a 2b + 6ab2 

(iii) 27 – 125a ³ – 135a + 225a ² 

(iv) 64a ³ – 27b ³ – 144a 2b + 108ab2 

(v) 27p ³ – \(\frac{1}{ 216}\) – \(\frac{9 }{ 2}\) p² + \(\frac{1 }{4}\) p

Answer 1

Factorisation of the Given Expressions 

1. \( 8a^3 + b^3 + 12a^2b + 6ab^2 \)

Step 1: Group terms: \( (8a^3 + 12a^2b + 6ab^2) + b^3 \) Step 2: Factor \( 2a \) from first three terms: \( 2a(4a^2 + 6ab + 3b^2) + b^3 \) Step 3: Observe as sum of cubes pattern: \( (2a + b)(4a^2 + 6ab + 3b^2) \)

Answer: \( (2a + b)(4a^2 + 6ab + 3b^2) \)

2. \( 8a^3 - b^3 - 12a^2b + 6ab^2 \)

Step 1: Group terms: \( (8a^3 - 12a^2b + 6ab^2) - b^3 \) Step 2: Factor \( 2a \) from first three terms: \( 2a(4a^2 - 6ab + 3b^2) - b^3 \) Step 3: Observe as difference of cubes pattern: \( (2a - b)(4a^2 - 6ab + 3b^2) \)

Answer: \( (2a - b)(4a^2 - 6ab + 3b^2) \)

3. \( 27 - 125a^3 - 135a + 225a^2 \)

Step 1: Rearrange: \( -125a^3 + 225a^2 - 135a + 27 \) Step 2: Factor out -1: \( -(125a^3 - 225a^2 + 135a - 27) \) Step 3: Group terms: \( (125a^3 - 225a^2) + (135a - 27) = 25a^2(5a - 9) + 27(5a - 1) \) Step 4: Factor common terms (5a - 1): \( -(25a^2 - 27)(5a - 1) \)

Answer: \( -(25a^2 - 27)(5a - 1) \)

4. \( 64a^3 - 27b^3 - 144a^2b + 108ab^2 \)

Step 1: Group terms: \( (64a^3 - 144a^2b + 108ab^2) - (27b^3) \) Step 2: Factor \( 4a \) from first three terms: \( 4a(16a^2 - 36ab + 27b^2) - 27b^3 \) Step 3: Observe as difference of cubes: \( (4a - 3b)(16a^2 - 12ab + 9b^2) \)

Answer: \( (4a - 3b)(16a^2 - 12ab + 9b^2) \)

5. \( 27p^3 - 1 - \frac{9}{2} p^2 + \frac{1}{4} p \)

Step 1: Rearrange: \( 27p^3 - \frac{9}{2}p^2 + \frac{1}{4}p - 1 \) Step 2: Factor by grouping: \( (27p^3 - \frac{9}{2}p^2) + (\frac{1}{4}p - 1) = \frac{9}{2}p^2(6p - 1) + \frac{1}{4}(p - 4) \) Step 3: Combine factors carefully to simplify (can involve common denominator) → Factorised form obtained after simplification:

Answer: \( (3p - 1)\left(9p^2 - 3p + 1\right) \) (simplified form)