Question

please solve the questions in the attachment. (asap)​

Answer 1

Question :-

Simplify

1. (a+3b)³ + (a-3b)³
2. (a-b)³ + (b-c)³ + (c-a)³

Factories

1. 64p³ + 27q³ + 144p²q + 108pq²
2. 8a³ - 27b³ - 36a²b + 54ab²
3. 125p³ - 27q³
4. 343x³ + 64y³

Solution :-

Simplify

1. (a+3b)³ + (a-3b)³

By applying identity → a³+b³ = (a+b)(a²-ab+b²)

= (a+3b+a-3b) ((a+3b)² - (a+3b)(a-3b) + (a-3b)²)

Applying Identities → (a+b)² = a²+b²+2ab

Applying Identities → (a+b)² = a²+b²+2ab(a-b)² = a²+b²-2ab and a²-b² = (a+b)(a-b)

= 2a(a²+9b²+6ab - (a²-9b²) + a²+9b²-6ab)

= 2a(a²+a²+9b²+9b²-a²+9b²+6ab-6ab)

= 2a(2a²-a²+27b²)

= 2a(a²+27b²)

2. (a-b)³ + (b-c)³ + (c-a)³

if x = (a-b) y = (b-c) z = (c-a)

Consider

x + y + z = (a-b) + (b-c) + (c-a) = 0

=> x³ + y³ + z³ = 3xyz

=> (a-b)³ + (b-c)³ + (c-a)³ = 3(a-b)(b-c)(c-a)

Factories

1. 64p³ + 27q³ + 144p²q + 108pq²

Applying identity → (a+b)³= a³+b³+3ab(a+b)

= (4p)³ + (3q)³ + 3×4p×3q(4p+3q)

= (4p+3q)³

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

2. 8a³ - 27b³ - 36a²b + 54ab²

Applying identity → (a-b)³ = a³ - b³ -3ab(a-b)

= (2a)³ - (3b)³ - 3×2a×3b(2a-3b)

= (2a-3b)³

= (2a-3b)(2a-3b)(2a-3b)

3. 125p³ - 27q³

Applying identity → a³ - b³ = (a-b)(a²+ab+b²)

= (5p)³ - (3q)³

= (5p-3q)((5p)² + 5p×3q + (3q)²)

= (5p-3q)(25p²+15pq+9q²)

4. 343x³ + 64y³

Applying identity → a³+b³ = (a+b)(a²-ab+b²)

= (7x)³ + (4y)³

= (7x+4y)((7x)²-7x×4y+(4y)²)

= (7x+4y)(49x²-28xy+16y²)