f3 factorization

Identityused : v² - u² = (v - u)(v + u)


(1)
196 - (3m - n)²
= 14² - (3m - n)²
= [14 + (3m - n)] [14 - (3m - n)]
= (14 + 3m - n)(14 - 3m + n)


(2)
(x + y)² - (x - 2y)²
= [(x + y) - (x - 2y)] [(x + y) + (x - 2y)]
= (x + y - x + 2y)(x + y + x - 2y)
= 3y(2x - y)


(3)
81(2h - k)² - 121(h - k)²
= 9²(2h - k)² - 11²(h - k)²
= (18h - 9k)² - (11h - 11k)²
= [(18h - 9k) - (11h - 11k)] [(18h - 9k) + (11h - 11k)]
= (18h - 9k - 11h + 11k)(18h - 9k + 11h - 11k)
= (7h + 2k)(29h - 20k)


(4)
a²(b + 1)² -b²(a + 1)²
= (ab + a)² - (ab + b)²
= [(ab + a) - (ab + b)] [(ab + a) + (ab + b)]
= (ab + a - ab - b)(ab + a + ab + b)
= (a - b)(a + b + 2ab)


(5)
p - 3q - p² + 9q²
= (p - 3q) - (p² - 9q²)
= (p - 3q) - [p² - (3q²)]
= (p - 3q) - (p - 3q)(p + 3q)
= (p - 3q)[1 - (p + 3q)]
= (p - 3q)(1 - p - 3q)


(6)
8h² - 50k² + 2h - 5k
= (8h² - 50k²) + (2h - 5k)
= 2[(2h)² - (5k)²] + (2h - 5k)
= 2(2h - 5k)(2h + 5k) + (2h - 5k)
= (2h - 5k) [2(2h + 5k) + 1]
= (2h - 5k)(4h + 10k + 1)


(7)
a⁴ - 625
= (a²)² - 25²
= (a² - 25)(a² + 25)
= (a² - 5²)(a² + 25)
= (a - 5)(a + 5)(a² + 25)


(8)
81y⁴ - 16z⁴
= (9y²)² - (4z²)²
= (9y² - 4z²)(9y² + 4z²)
= [(3y)² - (2z)²](9y² + 4z²)
= (3y - 2z)(3y + 2z)(9y² + 4z²)