Factorise the following:
x2 + xy - x - y
Answer
x2 + xy - x - y
= x(x + y) - 1(x + y)
= (x + y)(x - 1).
Hence, x2 + xy - x - y = (x - 1)(x + y).
Factorise the following:
y2 - yz - 5y + 5z
Answer
y2 - yz - 5y + 5z
= y(y - z) - 5(y - z)
= (y - z)(y - 5).
Hence, y2 - yz - 5y + 5z = (y - z)(y - 5)
Factorise the following:
5xy + 7y - 5y2 - 7x
Answer
5xy + 7y - 5y2 - 7x
Rearranging the terms:
= 5xy - 7x + 7y - 5y2
= x(5y - 7) - y(5y - 7)
= (x - y)(5y - 7)
Hence, 5xy + 7y - 5y2 - 7x = (x - y)(5y - 7).
Factorise the following:
5p2 - 8pq - 10p + 16q
Answer
5p2 - 8pq - 10p + 16q
= 5p2 - 10p - 8pq + 16q
= 5p(p - 2) - 8q(p - 2)
= (5p - 8q)(p - 2).
Hence, 5p2 - 8pq - 10p + 16q = (5p - 8q)(p - 2)
Factorise the following:
a2b - ab2 + 3a - 3b
Answer
a2b - ab2 + 3a - 3b
= ab(a - b) + 3(a - b)
= (ab + 3)(a - b).
Hence, a2b - ab2 + 3a - 3b = (ab + 3)(a - b).
Factorise the following:
x3 - 3x2 + x - 3
Answer
x3 - 3x2 + x - 3
= x2(x - 3) + 1(x - 3)
= (x2 + 1)(x - 3).
Hence, x3 - 3x2 + x - 3 = (x2 + 1)(x - 3).
Factorise the following:
6xy2 - 3xy - 10y + 5
Answer
6xy2 - 3xy - 10y + 5
= 3xy(2y - 1) - 5(2y - 1)
= (2y - 1)(3xy - 5).
Hence, 6xy2 - 3xy - 10y + 5 = (2y - 1)(3xy - 5).
Factorise the following:
3ax - 6ay - 8by + 4bx
Answer
3ax - 6ay - 8by + 4bx
= 3ax - 6ay + 4bx - 8by
Taking out common terms we get,
3ax - 6ay + 4bx - 8by
= 3a(x - 2y) + 4b(x - 2y)
= (x - 2y)(3a + 4b).
Hence, 3ax - 6ay - 8by + 4bx = (x - 2y)(3a + 4b).
5px - 8qy + 4qx - 10py
Answer
Given,
⇒ 5px - 8qy + 4qx - 10py
⇒ 5px + 4qx - 8qy - 10py
⇒ x(5p + 4q) - 2y(4q + 5p)
⇒ (5p + 4q)(x - 2y).
Hence, 5px - 8qy + 4qx - 10py = (5p + 4q)(x - 2y).
9a2y - 9ay + 6a - 6
Answer
Given,
⇒ 9a2y - 9ay + 6a - 6
⇒ 9ay(a - 1) + 6(a - 1)
⇒ (a - 1)(9ay + 6)
⇒ (a - 1).3.(3ay + 2)
⇒ 3(a - 1)(3ay + 2).
Hence, 9a2y - 9ay + 6a - 6 = 3(a - 1)(3ay + 2).
Factorise the following:
1 - a - b + ab
Answer
1 - a - b + ab
= 1(1 - a) - b(1 - a)
= (1 - a)(1 - b).
Hence, 1 - a - b + ab = (1 - a)(1 - b).
Factorise the following:
a(a - 2b - c) + 2bc
Answer
a(a - 2b - c) + 2bc
= a2 - 2ab - ac + 2bc
= a2 - ac - 2ab + 2bc
= a(a - c) - 2b(a - c)
= (a - 2b)(a - c).
Hence, a(a - 2b - c) + 2bc = (a - 2b)(a - c).
Factorise the following:
x2 + xy(1 + y) + y3
Answer
x2 + xy(1 + y) + y3
= x2 + xy + xy2 + y3
= x(x + y) + y2(x + y)
= (x + y)(x + y2).
Hence, x2 + xy(1 + y) + y3 = (x + y)(x + y2).
Factorise the following:
y2 - xy(1 - x) - x3
Answer
y2 - xy(1 - x) - x3
= y2 - xy + x2y - x3
= y(y - x) + x2(y - x)
= (y - x)(y + x2).
Hence, y2 - xy(1 - x) - x3 = (y - x)(y + x2).
Factorise the following:
ab2 + (a - 1)b - 1
Answer
ab2 + (a - 1)b - 1
= ab2 + ab - b - 1
= ab(b + 1) - 1(b + 1)
= (b + 1)(ab - 1)
Hence, ab2 + (a - 1)b - 1 = (b + 1)(ab - 1).
Factorise the following:
2a - 4b - xa + 2bx
Answer
2a - 4b - xa + 2bx
= 2(a - 2b) - x(a - 2b)
= (2 - x)(a - 2b).
Hence, 2a - 4b - xa + 2bx = (2 - x)(a - 2b).
Factorise the following:
5ph - 10qk + 2rph - 4qrk
Answer
5ph - 10qk + 2rph - 4qrk
= 5(ph - 2qk) + 2r(ph - 2qk)
= (5 + 2r)(ph - 2qk).
Hence, 5ph - 10qk + 2rph - 4qrk = (5 + 2r)(ph - 2qk).
Factorise the following:
x2 - x(a + 2b) + 2ab
Answer
x2 - x(a + 2b) + 2ab
= x2 - xa - 2bx + 2ab
= x(x - a) - 2b(x - a)
= (x - a)(x - 2b).
Hence, x2 - x(a + 2b) + 2ab = (x - a)(x - 2b).
Factorise the following:
ab(x2 + y2) - xy(a2 + b2)
Answer
ab(x2 + y2) - xy(a2 + b2)
= abx2 + aby2 - xya2 - xyb2
= abx2 - xyb2 - xya2 + aby2
= bx(ax - by) - ay(ax - by)
= (bx - ay)(ax - by).
Hence, ab(x2 + y2) - xy(a2 + b2) = (bx - ay)(ax - by).
Factorise the following:
(ax + by)2 + (bx - ay)2.
Answer
(ax + by)2 + (bx - ay)2
= a2x2 + b2y2 + 2axby + b2x2 + a2y2 - 2axby
= a2x2 + b2x2 + b2y2 + a2y2
= x2(a2 + b2) + y2(b2 + a2)
= (x2 + y2)(a2 + b2).
Hence, (ax + by)2 + (bx - ay)2 = (x2 + y2)(a2 + b2).
Factorise the following:
a3 + ab(1 - 2a) - 2b2.
Answer
a3 + ab(1 - 2a) - 2b2
= a3 + ab - 2a2b - 2b2
= a3 - 2a2b + ab - 2b2
= a2(a - 2b) + b(a - 2b)
= (a2 + b)(a - 2b).
a3 + ab(1 - 2a) - 2b2 = (a2 + b)(a - 2b).
Factorise the following:
3x2y - 3xy + 12x - 12.
Answer
3x2y - 3xy + 12x - 12.
= 3(x2y - xy + 4x - 4)
= 3[xy(x - 1) + 4(x - 1)]
= 3(x - 1)(xy + 4)
Hence, 3x2y - 3xy + 12x - 12 = 3(x - 1)(xy + 4).
Factorise the following:
a2b + ab2 - abc - b2c + axy + bxy.
Answer
a2b + ab2 - abc - b2c + axy + bxy
= ab(a + b) - bc(a + b) + xy(a + b)
= (a + b)(ab - bc + xy).
Hence, a2b + ab2 - abc - b2c + axy + bxy = (a + b)(ab - bc + xy).
Factorise the following:
ax2 - bx2 + ay2 - by2 + az2 - bz2.
Answer
ax2 - bx2 + ay2 - by2 + az2 - bz2
= x2(a - b) + y2(a - b) + z2(a - b)
= (a - b)(x2 + y2 + z2).
Hence, ax2 - bx2 + ay2 - by2 + az2 - bz2 = (a - b)(x2 + y2 + z2).
Factorise the following:
x - 1 - (x - 1)2 + ax - a.
Answer
x - 1 - (x - 1)2 + ax - a
= (x - 1) - (x - 1)2 + a(x - 1)
= (x - 1)[1 - (x - 1) + a]
= (x - 1)(1 - x + 1 + a)
= (x - 1)(a - x + 2).
Hence, x - 1 - (x - 1)2 + ax - a = (x - 1)(a - x + 2).