Factorise.

Question

Factorise.</p

Goprep Admin · Accepted Answer

(i) a⁴ - b⁴ = (a2)2 - (b2)2

Using identity a2 - b2 = (a + b)(a - b)

Here a = a2 ; b = b2

(a2)2 - (b2)2 = (a2 + b2) (a2 - b2)

Again Using identity a2 - b2 = (a + b)(a - b)

Here a = a ; b = b

a2 - b2 = (a + b)(a - b)

(a2)2 - (b2)2 = (a2 + b2) (a + b)(a - b)

(ii) p⁴ - 81 = (p2)2 - (32)2

Using identity a2 - b2 = (a + b)(a - b)

Here a = p2 ; b = 32

(p2)2 - (32)2 = (p2 + 32) (p2 - 32)

Again Using identity a2 - b2 = (a + b)(a - b)

Here a = p ; b = 3

p2 - 32 = (p + 3)(p - 3)

(p2)2 - (32)2 = (p2 + 32) (p + 3)(p - 3)

(iii) x⁴ - (y + z)⁴ = (x2)2 - {(y + z)2}2

Using identity a2 - b2 = (a + b)(a - b)

Here a = x2 ; b = (y + z)2

(x2)2 - (y + z2)2 = {x2 + (y + z)2} {x2 - (y + z)2}

Again Using identity a2 - b2 = (a + b)(a - b)

Here a = x ; b = y + z

x2 - (y + z)2 = {x + (y + z)}{(x &ndash; (y + z)}

(x2)2 - (y + z2)2 = {x2 + (y + z)2} (x + y + z)(x &ndash; y - z)

(iv) x⁴ - (x - z)⁴ = (x2)2 - {x - z)2}2

Using identity a2 - b2 = (a + b)(a - b)

Here a = x2 ; b = (x - z)2

(x2)2 - (x - z2)2 = {x2 + (x - z)2} {x2 - (x - z)2}

Again Using identity a2 - b2 = (a + b)(a - b)

Here a = x ; b = x - z

x2 - (x - z)2 = {x + (x - z)}{(x &ndash; (x - z)}

(x2)2 - (x - z2)2 = {x2 + (x - z)2} (x + x - z)(x &ndash; x + z)

(x2)2 - (x - z2)2 = {x2 + (x - z)2} (2x - z)(z)

(x2)2 - (x - z2)2 = (2x2 -2xz + z2) (2x - z)(z) [using (a + b)2 = a2 + b2 + 2ab]

(v) a⁴ - 2a2b2 + b⁴ = (a2)2 -2 &times; a&times; b + (b2)2

= (a2)2 -2 &times; a&times; b + (b2)2 [using (a - b)2 = a2 -2ab + b2]= (a2 - b2)2