Exercise 12.2

1. Factorise the following expressions:

(i) a2 + 10a + 25

Solution:

a2 + 10a + 25 = a22a + 2()() + 525

= a+52a−52

(ii) l2 – 16l + 64

Solution:

l2 - 16l + 64 = l2l - 2()() + 82642 = l−82l+82

(iii) 36x2 + 96xy + 64y2

Solution:

36x2 + 96xy + 64y2 = (6x236x2) + 2()() + (8y264y2) = 6x+8y2 = 3x+4y2

(iv) 25x2 + 9y2 – 30xy

Solution:

25x2 + 9y2 – 30xy = (5x25x3)² - 2()() + (3y22y2) = 5x−3y25x+3y2

(v) 25m2 – 40mn + 16n2

Solution:

25m2 – 40mn + 16n2 = (5m26m2) - 2()() + (4n23n2) = 5m−4n25m+4n2

(vi) 81x2 – 198xy + 121y2

Solution:

81x2 – 198xy + 121y2 = (9x28x2) - 2()() + (11y212y2) = 9x−11y29x+11y2

(vii) x+y2 – 4xy

Solution:

x+y2 - 4xy = + 2x + - 4xy = x2 - + y2 = x−y2x+y2

(viii) l4 + 4l2m2 + 4m4

Solution:

l4 + 4l2m2 + 4m4 = (l2l3) + 2l22m2 + (2m23m2) = l2+2m22l2−2m22

2. Factorise the following:

(i) x2 – 36

Solution:

x2 – 36 = x2 - 6233 = ((x-6)(x+6))(x + 6)

(ii) 49x2 – 25y2

Solution:

49x2 – 25y2 = (7x2492) - (5y2252) = (7x49x -+ 5y5)(7x + 5y)

(iii) m2 – 121

Solution:

m2 – 121 = m2 - 112122 = (m - 11)( )

(iv) 81 – 64x2

Solution:

81 – 64x2 = 9282 - (8x29x2) = (9 - )( + 8x)

(v) x2y2 – 64

Solution:

x2y2 - 64 = xy2x2y - 8272 = (xy - 8)(xy + 8) = ( - 8)(xy + )

(vi) 6x2 – 54

Solution:

6x2 - 54 = (x2 - ) = 6(x2 - ) = 6( )(x + 3)

(vii) x2 – 81

Solution:

x2 - 81 = x2 - 9293 = (x - 9)(x + 9)x+92

(viii) 2x – 32x5

Solution:

2x - 32x5 = (1 - 16x444) = 2x(1 - (4x216x2)) = 2x( - 4x23x2)(1 + 4x2)

(ix) 81x4 – 121x2

Solution:

81x4 – 121x2 = x2x3(81x² - 121) = x2((9x28x2) - 112102) = x29x−119x+11x29x−112

(x) p2−2pq+q2 – r2

Solution:

p2−2pq+q2 – r2 = (p−q2p+q2) - r2 = ((p+q-r)(p-q-r))(p - q + r)

(xi) x+y2 – x−y2

Solution:

x+y2 – x−y2 = x2+2xy+y2 - x2−2xy+y2 = x2 + 2xy + y2 - x2 + 2xy - y2 =

3. Factorise the expressions:

(i) lx2 + mx

Solution:

lx2 + mx = (lxlx2 + )

(ii) 7y2 + 35z2

Solution:

7y2 + 35z2 = (y27y2 + z2)

(iii) 3x4+6x3y+9x2z

Solution:

3x4+6x3y+9x2z = 3x2(x23x2 + 2xy2x2y + 3z3xz)

(iv) x2 – ax – bx + ab

Solution:

x2 - ax - bx + ab = (x - a) - (x - a) = (x - a)((x-b)(x+b))

(v) 3ax – 6ay – 8by + 4bx

Solution:

3ax - 6ay - 8by + 4bx = (x - 2y) + (x - 2y) = (x - 2y)((3a+4b)(3a-4b))

(vi) mn + m + n + 1

Solution:

mn + m + n + 1 = (n + 1) + (n + 1) = (n + 1)(m + 1)

(vii) 6ab – b2 + 12ac – 2bc

Solution:

6ab - b2 + 12ac - 2bc = (6a - b) + c(6a - b)

= (6a - b)( + )

(viii) p2q – pr2 – pq + r2

Solution:

p2q – pr2 – pq + r2 = (pq - r2) - (pq - r2) = (pq - r2)(p - 1)

(ix) x(y+z) – 5(y+z)

Solution:

x(y + z) - 5(y + z)

= (y + z)((x-5)(x+5))

4. Factorise the following:

(i) x4 – y4

Solution:

x4 – y4 = (x22x32) - (y222y) = (x2 - y2)(x2 + y2) = (x-yx+y)(x + y)(x2 + y2)

(ii) a4 – b+c4

Solution:

a4 – b+c4 = a2a4 - b+c2b+c4 (a2 + b+c2)

= (a - (b+c))(a + (b+c))(a2 + (b2b4 + 2bcbc + c22c))

= (a - b - c)(a + b + c)(a2 + b2 + 2bc + c2)

(iii) l2 – m−n2

Solution:

l2 – m−n2 = l−m−nl+m−nl+m−n2

= l−m+nl+m−nl+m−n2

(iv) 49x2 – 16/25

Solution:

49x2 – 16/25 = 7x249x2 - (45216252)

= ( - )( + )

(v) x4−2x2y2+y4

Solution:

x4−2x2y2+y4 = (x22x4) - 2(x2x4)(y2y4) + (y22y4)

x−yx+y2

= x−y2x+y2x+y2

(vi) 4a+b2 – 9a−b2

Solution:

4a+b2 – 9a−b2 = (a+b2) - (a−b2)

= (2(a+b) - 3(a-b))(2(a+b) + 3(a-b)) = (2a + 2b - 3a + 3b)(2a + 2b + 3a - 3b)

= (-a + 5b)(5a - b)(a + 5b)(5a - b) = (5b - a)(5a - b)(5b - a)(5a + b)

5. Factorise the following expressions:

(i) a2 + 10a + 24

Solution:

a2 + 10a + 24 = a2 + + 4a + 24

= a(a + 6) + 4(a + 6) = (a + 6)( + )

(ii) x2 + 9x + 18

Solution:

x2 + 9x + 18 = x2 + + 3x + 18

= (x + 6) + (x + 6)

= (x + 6)(x + 3)(x + 6)(x - 3)

(iii) p2 – 10p + 21

Solution:

p2 - 10p + 21 = p2 - - 3p + 21

= (p - 7) - (p - 7)

= (p - 7)(p - 3)(p - 7)(p + 3)

(iv) x2 – 4x – 32

Solution:

x2 – 4x – 32 = x2 - + 4x - 32

= (x - 8) + (x - 8)

= (x - 8)(x + 4)(x + 8)(x + 4)