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Mathematics Question on What is Factorisation?

Factorise

  1. a4−b4a^ 4 - b^ 4
  2. p4−81p^ 4 - 81
  3. x4−(y+z)4x^ 4 - (y + z)^ 4
  4. x4−(x−z)4x^ 4 - (x - z)^ 4
  5. a4−2a2b2+b4a ^4 - 2a ^2b^ 2 + b ^4
Answer

(i) a4−b4=(a2)2−(b2)2a^4 - b^4 = (a^2 )^2 - (b^ 2 )^ 2
= (a2−b2)(a2+b2)(a^ 2 - b^ 2 ) (a^ 2 + b ^2) )
= (a−b)(a+b)(a2+b2)(a - b) (a + b) (a^ 2 + b ^2 )

(ii) p4−81=(p2)2−(9)2p^ 4 - 81 = (p ^2 ) ^2 - (9)^2
= (p2−9)(p2+9)(p^ 2 - 9) (p ^2 + 9)
= [(p)2−(3)2](p2+9)[(p)^ 2 - (3)^2 ] (p ^2 + 9)
= (p−3)(p+3)(p2+9)(p - 3) (p + 3) (p^ 2 + 9)

(iii) x4−(y+z)4=(x2)2−[(y+z)2]2x^ 4 - (y + z) ^4 = (x ^2 )^ 2 - [(y +z) ^2 ] ^2
= [x2−(y+z)2][x2+(y+z)2][x^ 2 - (y + z) ^2 ] [x ^2 + (y + z) ^2 ]
= [x−(y+z)][x+(y+z)][x2+(y+z)2][x - (y + z)][ x + (y + z)] [x^ 2 + (y + z) ^2 ]
= (x−y−z)(x+y+z)[x2+(y+z)2](x - y - z) (x + y + z) [x^ 2 + (y + z) ^2 ]

(iv) x4−(x−z)4=(x2)2−[(x−z)2]2x^ 4 - (x - z)^ 4 = (x ^2 )^ 2 - [(x - z) ^2 ] ^2
= [x2−(x−z)2][x2+(x−z)2][x^ 2 - (x - z) ^2 ] [x ^2 + (x - z) ^2 ]
= [x−(x−z)][x+(x−z)][x2+(x−z)2][x - (x - z)] [x + (x - z)] [x^ 2 + (x - z) ^2 ]
= z(2x−z)[x2+x2−2xz+z2]z(2x - z) [x^ 2 + x ^2 - 2xz + z^ 2 ]
= z(2x−z)(2x2−2xz+z2)z(2x - z) (2x^ 2 - 2xz + z^ 2 )

(v) a4−2a2b2+b4=(a2)2−2(a2)(b2)+(b2)2a^ 4 - 2a 2b^ 2 + b ^4 = (a ^2 )^ 2 - 2 (a ^2 ) (b ^2 ) + (b ^2 ) ^2
= (a2−b2)2(a^ 2 - b^ 2 )^ 2
= [(a−b)(a+b)]2[(a - b) (a + b)]^2
= (a−b)2(a+b)2(a - b) ^2 (a + b) ^2