Question

Find the LCM and GCD for the following and verify that

f(x) x g(x) = LCM X GCD (x^3 - 1)(x+1) (X^3-1).​

Answer 1

Step-by-step explanation:

Find the LCM and GCD for the following and verify that f(x) x g(x) = LCM x GCD. (i) 21x2y, 35xy2 (ii) (x3 – 1)(x + 1), x3 + 1 (iii) (x3 – 1) (x + 1), (x3 – 1) (iv) (x2 y + xy2), (x2 + xy)Read more on Sarthaks.com - https://www.sarthaks.com/938791/find-the-lcm-and-gcd-for-the-following-and-verify-that-f-x-x-g-x-lcm-x-gcd-i-21x-2y-35xy-2-ii-x-3-1-x-1-x

Answer 2

• GCD = (x + 1)
• LCM = (x + 1) (x - 1) (x² + x + 1) (x² - x + 1)

Step-by-step explanation:

LHS,

(x³ - 1) = we can write it as (x³ - 1³).

⟼ (x³ - 1³) = (a³ - b³) = (a - b) (a² + ab + b²) (a + b)

⟼ (x³ - 1³) = (x - 1) (x² + x + 1) (x + 1)

RHS,

(x³ + 1) = we can write it as (x³ + 1³)

⟼ (x³ + 1³) = (a + b) (a² - ab + b²)

⟼ (x³ + 1³) = (x + 1) (x² - x + 1)

❶ GCD,

⟼ [(x - 1) (x² + x + 1) (x + 1)][(x + 1) (x² - x + 1)]

⟼ (x - 1) (x² + x + 1) (x + 1) (x + 1) (x² - x + 1)

⟼ GCD = (x + 1)

❷ LCM,

⟼ [(x - 1) (x² + x + 1) (x + 1)][(x + 1) (x² - x + 1)]

⟼ (x - 1) (x² + x + 1) (x + 1) (x + 1) (x² - x + 1)

⟼ (x + 1) (x - 1) (x² + x + 1) (x² - x + 1)

f(x) × g(x) = LCM × GCD,

⟼ [(x³ - 1) (x + 1)] × [(x³ + 1)] = [(x + 1) (x - 1) (x² + x + 1) (x² - x + 1)] × [(x + 1)]

⟼ [(x³ - 1) (x + 1)] × [(x³ + 1)] = [(x + 1) (x² - x + 1)] × [(x - 1) (x² + x + 1)] × [(x + 1)]

⟼ (x³ - 1) (x + 1) (x³ + 1) = (x³ - 1) (x³ + 1³) (x + 1)

⟼ (x³ - 1) (x + 1) (x³ + 1) = (x³ - 1) ( x + 1) (x³ + 1³)

• Hence Verified