EXPAND THE FOLLOWING USING SUITABLE IDENTIES

Expand the following :

Problem 1 :

(3m + 5)²

Solution :

= (3m + 5)²

Using the algebraic identity

(a + b)² = a² + 2ab + b²

Here a = 3m and b = 5

(3m + 5)² = (3m)² + 2(3m)(5) + 5²

 = 3²m² + 30m + 25

= 9m² + 30m + 25

Problem 2 :

(5p - 1)² 

Solution :

= (5p - 1)²

Using the algebraic identity

(a - b)² = a² - 2ab + b²

Here a = 5p and b = 1

(5p - 1)² = (5p)² - 2(5p)(1) + 1²

 = 5²p² - 10p + 1

= 25p² - 10p + 1

Problem 3 :

(2n - 1)(2n + 3)

Solution :

= (2n - 1)(2n + 3)

Multiplying the both terms.

= 4n² + 6n - 2n - 3

= 4n² + 4n - 3

Problem 4 :

4p² - 25q²

Solution :

= 4p² - 25q²

Using the algebraic identity

a² - b² = (a + b)(a - b)

4p² - 25q² = (2p)² - (5q)²

Here a = 2p and b = 5q

(2p)² - (5q)² = (2p + 5q)(2p - 5q)

Problem 5 :

(3 + m)³

Solution :

= (3 + m)³

Using the algebraic identity

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here a = 3 and b = m

(3 + m)³ = 3³ + 3(3)²(m) + 3(3)(m)² + m³

= 27 + 27m + 9m² + m³

= m³ + 9m² + 27m + 27

Problem 6  :

(2a + 5)³

Solution :

= (2a + 5)³

Using the algebraic identity

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here a = 2a and b = 5

(2a + 5)³ = (2a)³ + 3(2a)²(5) + 3(2a)(5)² + (5)³

= 8a³ + 60a² + 150a + 125

Problem 7 :

(3p + 4q)³

Solution :

= (3p + 4q)³

Using the algebraic identity

(a + b)³ = a³ + 3a²b + 3b²a + b³

Here a = 3p and b = 4q

(3p + 4q)³ = (3p)³ + 3(3p)²(4q) + 3(3p)(4q)² + (4q)³

= 27p³ + 108p²q + 144q²p + 64q³

Problem 8 :

(5 - x)³

Solution :

= (5 - x)³

Using the algebraic identity

(a - b)³ = a³ - 3a²b + 3b²a - b³

Here a = 5 and b = x

(5 - x)³ = (5)³ - 3(5)²x + 3(x)²(5) - (x)³

= 125 - 75x + 15x² - x³

Problem 9 :

(2x - 4y)³

Solution :

= (2x - 4y)³

Using the algebraic identity

(a - b)³ = a³ - 3a²b + 3b²a - b³

Here a = 2x and b = 4y

(2x - 4y)³ = (2x)³ - 3(2x)²(4y) + 3(4y)²(2x) - (4y)³

= 8x³ - 48x²y + 96xy² - 64y³

Problem 10 :

(ab - c)³

Solution :

= (ab - c)³

Using the algebraic identity

(a - b)³ = a³ - 3a²b + 3b²a - b³

Here a = ab and b = c

(ab - c)³ = (ab)³ - 3(ab)²(c) + 3(c)²(ab) - (c)³

= a³b³ - 3a²b²c + 3abc² - c³

Evaluate Using Algebraic Identities

Problem 1 :

(52)³

Solution :

= (50 + 2)³

Here a = 50 and b = 2

= 50³ + 3(50)² (2) + 3(50) 2² + 2³

= 125000 + 6(2500) + 12(50) + 8

= 125000 + 15000 + 600 + 8

= 140608

Problem 2 :

(104)³

Solution :

= (104)³

= (100 + 4)³

Here a = 100 and b = 4

= (100)³ + 3(100)²(4) + 3(100) 4² + 4³

= 1000000 + 120000 + 4800 + 64

= 1124864

Problem 3 :

(48)³

Solution :

= (48)³

= (50 - 2)³

a = 50 and b = 2

= 50³ - 3(50)² (2) + 3(50) 2² - 2³

= 125000 - 15000 + 600 - 8

= 110592