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Question
Factorise
(i) a4− b4
(ii) p4− 81
(iii) x4− (y + z)4
(iv) x4− (x − z)4
(v) a4− 2a2b2 + b4
Factorise
(i) a4− b4
(ii) p4− 81
(iii) x4− (y + z)4
(iv) x4− (x − z)4
(v) a4− 2a2b2 + b4
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Solution
(i) a4− b4 = (a2)2 −(b2)2
=(a2 − b2) (a2+ b2)
=(a − b) (a + b) (a2+ b2)
(ii) p4− 81 = (p2)2 − (9)2
= (p2− 9) (p2 + 9)
=[(p)2 − (3)2] (p2+ 9)
= (p− 3) (p + 3) (p2 + 9)
(iii) x4− (y + z)4 = (x2)2− [(y +z)2]2
=[x2 − (y + z)2] [x2+ (y + z)2]
=[x − (y + z)][ x + (y + z)][x2 + (y + z)2]
=(x − y − z) (x + y +z) [x2 + (y + z)2]
(iv) x4− (x − z)4 = (x2)2− [(x − z)2]2
=[x2 − (x − z)2][x2 + (x − z)2]
=[x − (x − z)] [x + (x− z)] [x2 + (x − z)2]
=z(2x − z) [x2 + x2− 2xz + z2]
=z(2x − z) (2x2 −2xz + z2)
(v) a4− 2a2b2 + b4= (a2)2 − 2 (a2)(b2) + (b2)2
=(a2 − b2)2
=[(a − b) (a + b)]2
=(a − b)2 (a + b)2
(i) a4− b4 = (a2)2 −(b2)2
=(a2 − b2) (a2+ b2)
=(a − b) (a + b) (a2+ b2)
(ii) p4− 81 = (p2)2 − (9)2
= (p2− 9) (p2 + 9)
=[(p)2 − (3)2] (p2+ 9)
= (p− 3) (p + 3) (p2 + 9)
(iii) x4− (y + z)4 = (x2)2− [(y +z)2]2
=[x2 − (y + z)2] [x2+ (y + z)2]
=[x − (y + z)][ x + (y + z)][x2 + (y + z)2]
=(x − y − z) (x + y +z) [x2 + (y + z)2]
(iv) x4− (x − z)4 = (x2)2− [(x − z)2]2
=[x2 − (x − z)2][x2 + (x − z)2]
=[x − (x − z)] [x + (x− z)] [x2 + (x − z)2]
=z(2x − z) [x2 + x2− 2xz + z2]
=z(2x − z) (2x2 −2xz + z2)
(v) a4− 2a2b2 + b4= (a2)2 − 2 (a2)(b2) + (b2)2
=(a2 − b2)2
=[(a − b) (a + b)]2
=(a − b)2 (a + b)2
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