Factorise the following:i) 81x4 – 121x2ii) (p2 – 2pq + q2) - r2i

1.	Factorise the following:i) 81x4 – 121x2ii) (p2 – 2pq + q2) - r2iii) (x + y)2 – (x - y)2
Answer» i) 81x⁴ – 121x² = 81x⁴ – 121x² = x²(81² – 121) = x²[(9x)² – (11)²] = x²(9x + 11) (9x -11) ii) (p² – 2pq + q²) - r² = (p² – 2pq + q²) – r² = (p – q)² – (r)² [∵ p² – 2pq + q² = (p – q)²] = (p – q + r) (p – q – r) iii) (x + y)² – (x – y)² (x + y)² – (x – y)² It is in the form of a² – b² a = x + y, b = x - y ∴ a² – b² = (a + b)(a-b) = (x + y + x – y) [(x + y) - (x – y)] = 2x [x + y - x + y] = 2x x 2y = 4xy

Factorise the following:i) 81x4 – 121x2ii) (p2 – 2pq + q2) - r2iii) (x + y)2 – (x - y)2

Answer»

i) 81x⁴ – 121x²

= 81x⁴ – 121x²

= x²(81² – 121)

= x²[(9x)² – (11)²]

= x²(9x + 11) (9x -11)

ii) (p² – 2pq + q²) - r²

= (p² – 2pq + q²) – r²

= (p – q)² – (r)² [∵ p² – 2pq + q² = (p – q)²]

= (p – q + r) (p – q – r)

iii) (x + y)² – (x – y)²

(x + y)² – (x – y)²

It is in the form of a² – b²

a = x + y, b = x - y

∴ a² – b² = (a + b)(a-b)

= (x + y + x – y) [(x + y) - (x – y)]

= 2x [x + y - x + y]

= 2x x 2y = 4xy