Expand and simplify (x + 1)(x + 3)(x + 1).
Worked Solution & Answer
(x2 + 4x + 3)(x + 1) = x3 + 5x2 + 7x + 3 B1
Answer: x3 + 5x2 + 7x + 3
Triple brackets — Notes & Worked Examples Show / Hide
Explanation:
To expand three brackets (x + a)(x + b)(x + c), break the problem into two clear stages: first expand and simplify any two of the brackets into a single quadratic expression (x2 + px + q).
Next, multiply every term in your simplified quadratic by every term in the third bracket (giving 6 terms before simplifying), then collect like terms in order: x3, x2, x, and the number term.
Example 1: Expand and simplify (x + 3)(x − 2)(x + 5).
Solution:
Expand first two: (x + 3)(x − 2) = x2 + x − 6
Multiply by (x + 5): x(x2 + x − 6) + 5(x2 + x − 6)
= x3 + x2 − 6x + 5x2 + 5x − 30
Collect like terms:
Answer: x3 + 6x2 − x − 30
Example 2: Expand and simplify (2x − 1)(x + 4)(x − 3).
Solution:
Expand last two: (x + 4)(x − 3) = x2 + x − 12
Multiply by (2x − 1): 2x(x2 + x − 12) − 1(x2 + x − 12)
= 2x3 + 2x2 − 24x − x2 − x + 12
Collect like terms:
Answer: 2x3 + x2 − 25x + 12
Expand and simplify (x + 1)(x + 4)(x + 1).
Worked Solution & Answer
(x2 + 5x + 4)(x + 1) = x3 + 6x2 + 9x + 4 B1
Answer: x3 + 6x2 + 9x + 4
Expand and simplify (2x + 3)(x - 5)(x + 4)
Worked Solution & Answer
(2x + 3)(x² − x − 20) = 2x³ − 2x² − 40x + 3x² − 3x − 60 = 2x³ + x² − 43x − 60. M2 A1
Answer: 2x3 + x2 - 43x - 60
Expand and simplify (x - 2)(3x + 2)(2x + 3)
Worked Solution & Answer
(x – 2)(3x + 2) = 3x² – 4x – 4. (3x² – 4x – 4)(2x + 3) = 6x³ + 9x² – 8x² – 12x – 8x – 12 = 6x³ + x² – 20x – 12 M2 A1
Answer: 6x3 + x2 - 20x - 12
Expand and simplify (3x - 1)(2x + 3)(x - 5)
Worked Solution & Answer
(3x – 1)(2x + 3) = 6x² + 7x – 3. Multiplying by (x – 5): 6x³ – 30x² + 7x² – 35x – 3x + 15 = 6x³ – 23x² – 38x + 15. M2 A1
Answer: 6x3 - 23x2 - 38x + 15
Expand and simplify fully (x - 3)(x + 2)(x + 5)
Answer
Worked Solution & Answer
(x - 3)(x + 2) = x2 - x - 6 M1
(x2 - x - 6)(x + 5) = x(x2 - x - 6) + 5(x2 - x - 6) M1
= x3 - x2 - 6x + 5x2 - 5x - 30
= x3 + 4x2 - 11x - 30 A1
(x2 - x - 6)(x + 5) = x(x2 - x - 6) + 5(x2 - x - 6) M1
= x3 - x2 - 6x + 5x2 - 5x - 30
= x3 + 4x2 - 11x - 30 A1
Answer: Answer x3 + 4x2 - 11x - 30
(a) Simplify fully (3x⁵y⁶)⁴ (2) (b) Expand and simplify (x + 2)(x – 3)(x + 4) (3)
(a) Simplify fully (3x5y6)4 [2 marks]
(b) Expand and simplify (x + 2)(x - 3)(x + 4) [3 marks]
Worked Solution & Answer
(a)
3⁴ x²⁰ y²⁴ = 81x²⁰y²⁴ M1 A1
Answer (a): 81x20y24
(b)
(x² – x – 6)(x + 4) = x³ + 4x² – x² – 4x – 6x – 24 = x³ + 3x² – 10x – 24 M2 A1
Answer (b): x3 + 3x2 - 10x - 24
(a) Expand and simplify fully (x + 2)(2x + 3)(3x + 4). [3 marks]
19 (b) Use your answer from part (a) to work out 102 × 203 × 304. [2 marks]
19 (b) Use your answer from part (a) to work out 102 × 203 × 304. [2 marks]
(a) Expand and simplify fully (x + 2)(2x + 3)(3x + 4) [3 marks]
Answer
(b) Use your answer from part (a) to work out 102 × 203 × 304 [2 marks]
Answer
Worked Solution & Answer
(a)
(x + 2)(2x + 3) = 2x² + 7x + 6. M2
(2x² + 7x + 6)(3x + 4) = 6x³ + 8x² + 21x² + 28x + 18x + 24 = 6x³ + 29x² + 46x + 24 A1
(2x² + 7x + 6)(3x + 4) = 6x³ + 8x² + 21x² + 28x + 18x + 24 = 6x³ + 29x² + 46x + 24 A1
Answer (a): Answer 6x3 + 29x2 + 46x + 24
(b)
Let x = 100: 6(1 000 000) + 29(10 000) + 46(100) + 24 = 6 294 624 M1 A1
Answer (b): Answer 6 294 624
(a) Expand and simplify 5x(x + 2)(3x – 4) (3) (b) Simplify completely (16w⁸ / y²⁰) {}-3/4) (3)
(a) Expand and simplify 5x(x + 2)(3x - 4) [3 marks]
(b) Simplify completely (16w8y20)-<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">4</span></span> [3 marks]
Worked Solution & Answer
(a)
5x(3x² + 2x – 8) = 15x³ + 10x² – 40x M2 A1
Answer (a): 15x3 + 10x2 - 40x
(b)
(y²⁰ / 16w⁸) {}3/4) = (y⁵ / 2w²)³ = y¹⁵ / 8w⁶ M2 A1
Answer (b): y158w6
(a) Expand and simplify (3x + 1)(2 − x)(4 + x) (3) (b) Simplify fully ((a³b)/(a⁹b⁵)) {}-1/2) (3)
(a) Expand and simplify (3x + 1)(2 - x)(4 + x) [3 marks]
(b) Simplify fully (a3ba9b5)-<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> [3 marks]
Worked Solution & Answer
(a)
(3x + 1)(2 − x) = −3x² + 5x + 2 M2
(−3x² + 5x + 2)(4 + x) = −12x² − 3x³ + 20x + 5x² + 8 + 2x = −3x³ − 7x² + 22x + 8 A1
(−3x² + 5x + 2)(4 + x) = −12x² − 3x³ + 20x + 5x² + 8 + 2x = −3x³ − 7x² + 22x + 8 A1
Answer (a): -3x3 - 7x2 + 22x + 8
(b)
Inside bracket = a⁻⁶b⁻⁴ M2
(a⁻⁶b⁻⁴) {}-1/2) = a³b² A1
(a⁻⁶b⁻⁴) {}-1/2) = a³b² A1
Answer (b): a3b2