# Factorize each of the following expressions :

### (1) qr – pr + qs – ps

#### Solution:

qr – pr + qs – ps

= (qr – pr) + (qs – ps)

= r(q – p) + s(q – p)

(q – p) as the common factor

= (r + s) (q – p)

### (2) p2q – pr2 – pq + r2

#### Solution:

p2q – pr2 – pq + r2

= (p2q – pq) + (r2 – pr2)

= pq(p – 1) + r2(1 – p)

= pq(p – 1) – r2(p – 1)

(p – 1) as the common factor

= (pq – r2)(p – 1)

### (3) 1 + x + xy + x2y

#### Solution:

1 + x + xy + x2y

= (1 + x) + (xy + x2y)

= (1 + x) + xy(1 + x)

(1 +x) as the common factor

= (1 + xy)(1 +x)

### (4) ax + ay – bx – by

#### Solution:

ax + ay – bx – by

= (ax + ay) – (bx + by)

= a(x + y) – b(x + y)

(x + y) as the common factor

= (a – b)(x + y)

### (5) xa2 + xb2 – ya2 – yb2

#### Solution:

xa2 + xb2 – ya2 – yb2

= (xa2 + xb2 ) – (ya2 + yb2 )

= x(a2 + b2) – y(a2 + b2)

(a2 + b2) as the common factor

= (x – y)(a2 + b2)

### (6) x2 + xy +xz + yz

#### Solution:

x2 + xy +xz + yz

= (x2 + xy) + (xz + yz)

= x(x + y) + z(x + y)

Here (x + y) is the common factor

= (x + z)(x + y)

= (x +y)(x +z)

### (7) 2ax + bx + 2ay + by

#### Solution:

2ax + bx + 2ay + by

= (2ax + bx) + (2ay + by)

= x(2a + b) + y(2a + b)

Here (2a + b) as the common factor

= (x + y)(2a + b)

### (8) ab – by – ay + y2

#### Solution:

ab – by – ay + y2

= (ab – ay) + (y2 – by)

= a(b – y) + y(y – b)

Here (b – y) as the common factor

= a(b – y) – y(b – y)

= (a – y)(b – y)

### (9) axy + bcxy – az – bcz

#### Solution:

axy + bcxy – az – bcz

= (axy + bcxy) – (az – bcz)

= xy(a + bc) – z(a + bc)

= (xy – z)(a + bc) [taking (a + bc) as the common factor]

### (10) lm2 – mn2– Lm + n2

#### Solution:

lm2 – mn2– lm + n2

= (lm2 – lm) + (n2 – mn2)

= lm(m – 1) + n2(1 – m)

= lm(m – 1) – n2(m – 1)

Here (m – 1) a sthe common factor

= (lm – n2)(m – 1)

### (11) x3 – y2 + x – x2y2

#### Solution:

x3 – y2 + x – x2y2

= (x3 + x) – (x2y2 + y2)

= x(x2 + 1) – y2(x2 + 1)

Here (x2+ 1) as the common factor

= (x – y2)(x2 + 1)

### (12) 6xy + 6 – 9y – 4x

#### Solution:

6xy + 6 – 9y – 4x

= (6xy – 4x) + (6 – 9y)

= 2x (3y -2) + 3(2 – 3y)

= 2x(3y – 2) – 3(3y – 2)

= (2x – 3)(3y -2)

### (13) x2 – 2ax – 2ab + bx

#### Solution:

x2 – 2ax – 2ab + bx

= (x2 – 2ax) + (bx – 2ab)

= x(x – 2a) + b(x – 2a)

= (x + b)(x – 2a)

#### Solution:

x3 – 2x2y + 3xy2 – 6y3

= (x3 – 2x2y) + (3xy2 – 6y3)

= x2(x – 2y) + 3y2(x – 2y)

= (x2 + 3y2)(x – 2y)

#### Solution:

abx2 + (ay – b)x – y

= (abx2 – bx) + (axy – y)

= bx (ax – 1) + y(ax – 1)

= (bx + y)(ax – 1)

### (16) (ax + by)2 + (bx – ay)2

#### Solution:

(ax + by)2 + (bx – ay)2

(using identities: (a + b)^2 = a2 + b2 + 2ab ) and (a – b)^2 = a2 + b2 – 2ab ) )

= a2x2 + 2abxy + b2y2 + b2x2 – 2abxy + a2y2

= (a2x2 + a2y2) + (b2x2 + b2y2)

= a(x+ y2) + b2(x2 + y2)

= (a 2 + b2)(x2 + y2)

### (17) 16(a – b)3 – 24(a – b)2

#### Solution:

16(a – b)3 – 24(a – b)2

= 8(a – b)2 [2(a – b) – 3]

= 8(a – b)2(2a – 2b – 3)

### (18) ab(x2 + 1) + x(a2 + b2)

#### Solution:

ab(x2 + 1) + x(a2 + b2)

= abx2 + ab + a2x + b2x

= (abx2 + a2x) + (b2x + ab)

= ax(bx + a) + b(bx + a)

= (ax + b)(bx + a)

### (19) a2x2 + (ax2 + 1)x + 1 + a

#### Solution:

a2x2 + (ax2 + 1)x + 1 + a

= a2x2 + ax3 + x + a

= ax2(x + a) + (x + a)

= (ax2 + 1)(x + a)

### (20) a(a – 2b – c) + 2bc

#### Solution:

a(a – 2b – c) + 2bc

= a2 – 2ab – ac + 2bc

= (a2 – ac) + (2bc – 2ab)

= a(a – c) + 2b(c – a)

= a(a – c) – 2b(a -c)

= (a – 2b)(a – c)

### (21) a(a + b – c) – bc

#### Solution:

a(a + b – c) – bc

= a2 + ab – ac – bc

= (a2 – ac) + (ab – bc)

= a(a – c) + b(a – c)

= (a + b)(a – c)

### (22) x2 – 11xy – x + 11y

#### Solution:

x2 – 11xy – x + 11y

= (x2 – x) + (11y – 11xy)

= x(x – 1) + 11y(1 – x)

= x(x – 1) – 11y(x – 1)

= (x – 11y)(x – 1)

### (23) ab – a – b + 1

#### Solution:

ab – a – b + 1

= (ab – b) + (1 – a)

= b(a – 1) + (1 – a)

= b(a – 1) – (a – 1)

= (a – 1)(b – 1)

### (24) x2 + y – xy – x

#### Solution:

x2 + y – xy – x

= (x2 – xy) + (y – x)

= x(x – y) + (y – x)

= x(x – y) – (x – y)

= (x – 1)(x – y)

Updated: November 12, 2019 — 3:57 pm