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Mathematics Question on What is Factorisation?

Factorise the following expressions.

  1. a2+8a+16a^ 2 + 8a + 16
  2. p2−10p+25p^ 2 - 10p + 25
  3. 25m2+30m+925m^2 + 30m + 9
  4. 49y2+84yz+36z249y^ 2 + 84yz + 36z ^2
  5. 4x2−8x+44x^ 2 - 8x + 4
  6. 121b2−88bc+16c2121b ^2 - 88bc + 16c ^2
  7. (l+m)2−4lm(l + m) ^2 - 4lm (Hint: Expand (l+m)2(l + m) ^2 first)
  8. a4+2a2b2+b4a^ 4 + 2a ^2b ^2 + b^ 4
Answer

(i) a2+8a+16=(a)2+2×a×4+(4)2a^ 2 + 8a + 16 = (a)^ 2 + 2 \times a \times 4 + (4)^2
=(a+4)2[(x+y)2=x2+2xy+y2] (a + 4)^2 [(x + y)^ 2 = x ^2 + 2xy + y^ 2 ]

(ii) p2−10p+25=(p)2−2×p×5+(5)2p^ 2 - 10p + 25 = (p)^ 2 - 2 \times p \times 5 + (5)^2
= (p−5)2[(a−b)2=a2−2ab+b2](p - 5)^2 [(a - b)^ 2 = a ^2 - 2ab + b ^2]

(iii) 25m2+30m+9=(5m)2+2×5m×3+(3)225m^2 + 30m + 9 = (5m) ^2 + 2 \times 5m \times 3 + (3)^2
= (5m+3)2[(a+b)2=a2+2ab+b2](5m + 3)^2 [(a + b) ^2 = a ^2 + 2ab + b^ 2 ]

(iv) 49y2+84yz+36z2=(7y)2+2×(7y)×(6z)+(6z)249y ^2 + 84yz + 36z^ 2 = (7y)^ 2 + 2 \times (7y) \times (6z) + (6z)^ 2
= (7y+6z)2[(a+b)2=a2+2ab+b2](7y + 6z)^ 2 [(a + b)^ 2 = a ^2 + 2ab + b^ 2 ]

(v) 4x2−8x+4=(2x)2−2(2x)(2)+(2)24x^ 2 - 8x + 4 = (2x)^ 2 - 2 (2x) (2) + (2)^2
= (2x−2)2[(a−b)2=a2−2ab+b2](2x - 2)^2 [(a - b)^ 2 = a^ 2 - 2ab + b ^2 ]
= [(2)(x−1)]2=4(x−1)2[(2) (x - 1)]^2 = 4(x - 1)^2

(vi) 121b2−88bc+16c2=(11b)2−2(11b)(4c)+(4c)2121b^ 2 - 88bc + 16c^ 2 = (11b)^ 2 - 2 (11b) (4c) + (4c)^ 2

= (11b−4c)2[(a−b)2=a2−2ab+b2](11b - 4c)^ 2 [(a - b)^ 2 = a ^2 - 2ab + b ^2 ]

(vii) (l+m)2−4lm=l2+2lm+m2−4lm(l + m)^ 2 - 4lm = l^ 2 + 2lm + m^2 - 4lm
=l2−2lm+m2l^ 2 - 2lm + m^2
= (l−m)2[(a−b)2=a2−2ab+b2](l - m)^ 2 [(a - b)^ 2 = a ^2 - 2ab + b^ 2 ]

(viii) 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+b)2=a2+2ab+b2](a^ 2 + b^ 2 )^ 2 [(a + b)^ 2 = a^ 2 + 2ab + b ^2 ]