Factorize each of the following expressions:
(a+b)³ -8(a-b)³
Answers
The algebraic expression is given as : (a + b)³ - 8(a - b)³
= (a + b)³ − [2(a − b)]³
= (a + b)³ − [2a − 2b] ³
In order to factorise the given algebraic expression as the differences of two cubes we use the following identities - a³ - b³ = (a - b) (a² + ab + b²) :
Here a = (a + b) and b = (2a − 2b) :
= (a + b − (2a − 2b)) ((a + b)² + (a + b)(2a − 2b) + (2a − 2b)²)
By using an identity , (a + b)² = a² + b² + 2ab :
=(a + b− 2a + 2b)(a² + b² + 2ab + (a + b)(2a − 2b) + (2a − 2b)²)
On multiplying the terms (a + b)(2a − 2b) :
= (a − 2a + b + 2b)(a² + b² + 2ab + 2a² −2ab + 2ab −2b² + (2a − 2b)²)
=(3b − a)(3a² + 2ab − b² + (2a − 2b)²)
By using an identity , (a + b)² = a² + b² + 2ab :
= (3b − a)(3a² + 2ab − b² + ((2a)² + (2b)² - 2 × 2a × 2b)
= (3b − a)(3a² + 2ab − b² + 4a² + 4b² − 8ab)
=(3b − a)(3a² + 4a² − b² + 4b² − 8ab + 2ab)
= (3b − a)(7a² + 3b² − 6ab)
= (- a + 3b)(7a² + 3b² − 6ab)
Hence, the factorization of an algebraic expression is (- a + 3b)(7a² + 3b² − 6ab) .
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Answer:
Step-by-step explanation:
= (a + b − (2a − 2b)) ((a + b)² + (a + b)(2a − 2b) + (2a − 2b)²)
=(a + b− 2a + 2b)(a² + b² + 2ab + (a + b)(2a − 2b) + (2a − 2b)²)= (a − 2a + b + 2b)(a² + b² + 2ab + 2a² −2ab + 2ab −2b² + (2a − 2b)²)
=(3b − a)(3a² + 2ab − b² + (2a − 2b)²)
= (3b − a)(3a² + 2ab − b² + ((2a)² + (2b)² - 2 × 2a × 2b)
= (3b − a)(3a² + 2ab − b² + 4a² + 4b² − 8ab)
=(3b − a)(3a² + 4a² − b² + 4b² − 8ab + 2ab)
= (3b − a)(7a² + 3b² − 6ab)
= (- a + 3b)(7a² + 3b² − 6ab)