1
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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