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Mathematics Question on What is Factorisation?

Factorise

  1. 4p2−9q24p^ 2 - 9q ^2
  2. 63a2−112b263a^ 2 - 112b ^2
  3. 49x2−3649x^ 2 - 36
  4. 16x5−144x316x^ 5 - 144x^ 3
  5. (l+m)2−(l−m)2(l + m)^ 2 - (l - m) ^2
  6. 9x2y2−169x^ 2 y^ 2 - 16
  7. (x2−2xy+y2)−z2(x^ 2 - 2xy + y^ 2 ) - z^ 2
  8. 25a2−4b2+28bc−49c225a ^2 - 4b^ 2 + 28bc - 49c ^2
Answer

(i) 4p2−9q2=(2p)2−(3q)24p^ 2 - 9q^ 2 = (2p) ^2 - (3q)^ 2

= (2p+3q)(2p−3q)[a2−b2(2p + 3q) (2p - 3q) [a ^2 - b ^2 = (a−b)(a+b)](a - b) (a + b)]

(ii) 63a2−112b263a^ 2 - 112b ^2

= 7(9a2−16b2)7(9a^ 2 - 16b^ 2 )

= 7[(3a)2−(4b)2]7[(3a)^2 - (4b) ^2 ]

= 7(3a+4b)(3a−4b)[a2−b2=(a−b)(a+b)]7(3a + 4b) (3a - 4b) [a^ 2 - b^ 2= (a - b) (a + b)]

(iii) 49x2−3649x ^2 - 36

= (7x)2−(6)2(7x)^ 2 - (6)^2

= (7x−6)(7x+6)[a2−b2=(a−b)(a+b)](7x - 6) (7x + 6) [a^ 2 - b^ 2 = (a - b) (a + b)]

(iv) 16x5−144x316x ^5 - 144x ^3

= 16x3(x2−9)16x^ 3 (x^ 2 - 9)

= 16x3[(x)2−(3)2]16 x^ 3 [(x)^ 2 - (3)^2 ]

= 16x3(x−3)(x+3)[a2−b2=(a−b)(a+b)]16 x^ 3 (x - 3) (x + 3) [a^ 2 - b ^2 = (a - b) (a + b)]

(v) (l+m)2−(l−m)2(l + m) ^2 - (l - m) ^2

= [(l+m)−(l−m)][(l+m)+(l−m)][Using  identity  a2−b2=(a−b)(a+b)][(l + m) - (l - m)] [(l + m) + (l - m)] [Using \;identity \;a ^2 - b ^2 = (a - b) (a + b)]

= (l+m−l+m)(l+m+l−m)(l + m - l + m) (l + m + l - m)

= 2mx2l=4ml=4lm2m x ^2l = 4ml = 4lm

(vi) 9x2y2−169x^ 2 y^ 2 - 16

= (3xy)2−(4)2(3xy) ^2 - (4)^2

= (3xy−4)(3xy+4)[a2−b2=(a−b)(a+b)](3xy - 4) (3xy + 4) [a ^2 - b ^2 = (a - b) (a + b)]

(vii) (x2−2xy+y2)−z2(x^ 2 - 2xy + y ^2 ) - z^ 2

= (x−y)2−(z)2[(a−b)2=a2−2ab+b2](x - y) ^2 - (z) ^2 [(a - b)^ 2 = a ^2 - 2ab + b ^2 ]

= (x−y−z)(x−y+z)[a2−b2=(a−b)(a+b)](x - y - z) (x - y + z) [a^ 2 - b^ 2 = (a - b) (a + b)]

(viii) 25a2−4b2+28bc−49c225a ^2 - 4b^ 2 + 28bc - 49c^ 2

= 25a2−(4b2−28bc+49c2)25a^ 2 - (4b^ 2 - 28bc + 49c ^2 )

= (5a)2−[(2b)2−2×2b×7c+(7c)2](5a)^ 2 - [(2b)^ 2 - 2 \times 2b \times 7c + (7c)^2 ]

= (5a)2−[(2b−7c)2][Using  identity(a−b)2=a2−2ab+b2](5a) ^2 - [(2b - 7c)^2 ] [Using \;identity (a - b) ^2 = a ^2 - 2ab + b ^2 ]

= [5a+(2b−7c)][5a−(2b−7c)][Using  identity  a2−b2=(a−b)(a+b)][5a + (2b - 7c)] [5a - (2b - 7c)] [Using \;identity \;a^ 2 - b ^2 = (a - b) (a + b)]

= (5a+2b−7c)(5a−2b+7c)(5a + 2b - 7c) (5a - 2b + 7c)