FACTORING POLYNOMIALS USING ALGEBRAIC IDENTITIES

Factorize each polynomial using algebraic identity.

Problem 1 :

36k² - 1

Solution :

36k² - 1 = (6k)² - 1²      

a² - b² = (a + b) (a - b)

(6k)² - 1² = (6k + 1) (6k - 1)

Problem 2 :

x² - 9y²

Solution :

x² - 9y² = x² - (3y)²

a² - b² = (a + b) (a - b)

x² - (3y)² = (x + 3y) (x - 3y)

Problem 3 :

25m² - n²

Solution :

25m² - n² = (5m)² - n²

a² - b² = (a + b) (a - b)

(5m)² - n² = (5m + n) (5m - n)

Problem 4 :

8r³ - 729

Solution :

8r³ - 729 = 8r³ - 9³

a³ - b³ = (a - b) (a² + ab + b²)

8r³ - 9³ = (8r - 9) [(8r)² + (8r)(9) + 9²]

= (8r - 9) (64r² + 72r + 81)

Problem 5 :

p³ - 1000q³

Solution :

p³ - 1000q³ = p³ - (10q)³

a³ - b³ = (a - b) (a² + ab + b²)

p³ - (10q)³ = (p - 10q) [(p)² + (p)(10q) + (10q)²]

= (p - 10q) (p² + 10pq + 100q²)

Problem 6 :

27c³ + 125d³

Solution :

27c³ + 125d³ = (3c)³ + (5d)³

a³ + b³ = (a + b) (a² - ab + b²)

(3c)³ + (5d)³ = (3c + 5d) [(3c)² - (3c)(5d) + (5d)²]

= (3c + 5d) (9c² - 15cd + 25d²)

Problem 7 :

64y³- 216

Solution :

64y³- 216 = (4y)³ - 6³

a³ - b³ = (a - b) (a² + ab + b²)

(4y)³ - 6³ = (4y - 6) [(4y)² + (4y)(6) + 6²]

= (4y - 6) (16y² + 24y + 36)

Problem 8 :

36y² + 84y + 49

Solution :

(a + b)² = a² + 2ab + b²

36y² + 84y + 49 = (6y)² + 2(6y)(7) + 7²

= (6y + 7)²

= (6y + 7) (6y + 7)

Problem 9 :

h² + 4h + 4

Solution :

(a + b)² = a² + 2ab + b²

h² + 4h + 4 = h² + 2(h)(2) + 2²

= (h + 2)²

= (h + 2) (h + 2)

Problem 10 :

z² - 8z + 16

Solution :

(a - b)² = a² - 2ab + b²

z² - 8z + 16 = z² - 2(z)(4) + 4²

= (z - 4)²

= (z - 4) (z - 4)

Problem 11 :

25a² - 9b²

Solution :

= 25a² - 9b²

= (5a)² - (3b)²

= (5a + 3b) (5a - 3b)

Problem 12 :

a² - 81(b - c)²

Solution :

= a² - 81(b - c)²

= a² - 9²(b - c)²

= a² - 9(b - c)²

= (a + 9(b - c)) (a - 9(b- c))

= (a + 9b - 9c) (a - 9b + 9c)

Problem 13 :

50a³ - 2a

Solution :

= 50a³ - 2a

= 2a (25a² - 1)

= 2a (5²a² - 1)

= 2a ((5a)² - 1²)

= 2a (5a + 1)(5a - 1)

Problem 14 :

3a⁵ - 108a³

Solution :

= 3a⁵ - 108a³

Factoring 3a³, we get

= 3a³ (a² - 36)

= 3a³ (a² - 6²)

= 3a³ (a + 6)(a - 6)

Problem 15 :

a⁴ - 1

Solution :

= a⁴ - 1

= (a²)² - (1²)²

= (a² + 1) (a² - 1)

= (a² + 1) (a² - 1²)

= (a² + 1) (a + 1)(a - 1)

Problem 16 :

(a + b)³ - a - b

Solution :

= (a + b)³ - a - b

= (a + b)(a² - ab + b²) - (a + b)

= (a + b)[a² - ab + b² - 1]

Problem 16 :

4a² - (4b² + 4 bc + c²)

Solution :

= 4a² - (4b² + 4 bc + c²)

= 4a² - (2²b² + 2(2b)c + c²)

= 4a² - [(2b)² + 2(2b)c + c²]

= 4a² - [2b + c]²

= 2²a² - [2b + c]²

= (2a)² - [2b + c]²

(2a + 2b + c) (2a - 2b - c)

Problem 17 :

a² - b² - (a + b)²

Solution :

= a² - b² - (a + b)²

= (a + b)(a - b) - (a + b)²

= (a + b)(a - b) - (a + b)(a + b)

Factoring (a + b), we get

= (a + b)(a - b - (a + b))

= (a + b)(a - b - a - b)

= (a + b)(-2 b)

= -2b(a + b)

Problem 18 :

a³ + 27b³

Solution :

= a³ + 27b³

= a³ + 3³b³

= a³ + (3b)³

= (a + 3b)(a² - ab + (3b)²)

= (a + 3b)(a² - ab + 9b²)