Chapter 7: Factorization Exercise – 7.4

Question: 1

Factorize of the following expressions:

qr – pr + qs – ps

Solution:

qr – pr + qs – ps

= (qr – pr) + (qs – ps)

= r(q – p) + s(q – p)

= (r + s) (q – p) [taking (q – p) as the common factor]

Question: 2

Factorize of the following expressions:

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) [since, (1 – p) = -(p – 1)]

= (pq – r²)(p – 1) [taking (p – 1) as the common factor]

Question: 3

Factorize of the following expressions:

1 + x + xy + x²y

Solution:

1 + x + xy + x²y

= (1 + x) + (xy + x²y)

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

= (1 + xy)(1 + x) [taking (1 + x) as the common factor]

Question: 4

Factorize of the following expressions:

ax + ay – bx – by

Solution:

ax + ay – bx – by

= (ax + ay) – (bx + by)

= a(x + y) – b(x + y)

= (a – b)(x + y) [taking (x + y) as the common factor]

Question: 5

Factorize of the following expressions:

xa² + xb² – ya² – yb²

Solution:

xa² + xb² – ya² – yb²

= (xa² + xb²) – (ya² + yb²)

= x(a² + b²) – y(a² + b²)

= (x – y)(a² + b²) [taking (a² + b²) as the common factor]

Question: 6

Factorize of the following expressions:

x² + xy + xz + yz

Solution:

x² + xy + xz + yz

= (x² + xy) + (xz + yz)

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

= (x + z)(x + y) [taking (x + y) s the common factor]

= (x + y)(x + z)

Question: 7

Factorize of the following expressions:

2ax + bx + 2ay + by

Solution:

2ax + bx + 2ay + by

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

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

= (x + y)(2a + b) [taking (2a + b) as the common factor]

Question: 8

Factorize of the following expressions:

ab – by – ay + y²

Solution:

ab – by – ay + y²

= (ab – ay) + (y² – by)

= a(b – y) + y(y – b) [since, (y – b) = – (b – y)]

= a(b – y) – y(b – y) [taking (b – y) as the common factor]

= (a – y)(b – y)

Question: 9

Factorize of the following expressions:

 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]

Question: 10

Factorize of the following expressions:

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) [since, (1 – m) = -(m – 1)]

= (Lm – n²)(m – 1) [taking (m – 1) as the common factor]

Question: 11

Factorize of the following expressions:

 x³ – y² + x – x²y²

Solution:

x³ - y² + x – x²y²

= (x³ + x) – (x²y² + y²)

= x(x² + 1) – y²(x² + 1)

= (x – y²)(x² + 1) [taking (x² + 1) as the common factor]

Question: 12

Factorize of the following expressions:

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) [since, (2 – 3y) = -(3y – 2)]

= (2x – 3)(3y – 2) [taking (3y – 2) as the common factor]

Question: 13

Factorize of the following expressions:

x² – 2ax – 2ab + bx

Solution:

x² – 2ax – 2ab + bx

= (x² – 2ax) + (bx – 2ab)

= x(x – 2a) + b(x – 2a)

= (x + b)(x – 2a) [taking (x – 2a) as the common factor]

= (x – 2a)(x + b)

Question: 14

Factorize of the following expressions:

x³ – 2x²y + 3xy² – 6y³

Solution:

x³ – 2x²y + 3xy² – 6y³

= (x³ – 2x²y) + (3xy² – 6y³)

= x²(x – 2y) + 3y²(x – 2y)

= (x² + 3y²)(x – 2y) [taking (x – 2y) as the common factor]

Question: 15

Factorize of the following expressions:

abx² + (ay – b)x – y

Solution:

abx² + (ay – b)x – y = abx² + axy – bx – y

= (abx² – bx) + (axy – y)

= bx (ax – 1) + y(ax – 1)

= (bx + y)(ax – 1) [taking (ax – 1) as the common factor]

Question: 16

Factorize of the following expressions:

(ax + by)² + (bx – ay)²

Solution:

(ax + by)² + (bx – ay)² = a²x² + 2abxy + b²y² + b²x² – 2abxy + a²y²

= a²x² + b²y² + b²x² + a²y²

= (a²x² + a²y²) + (b²x² + b²y²)

= a²(x² + y²) + b^2’(x² + y²)

= (a² + b²)(x² + y²) [taking (x² + y²) as the common factor]

Question: 17

Factorize of the following expressions:

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] [taking 8(a -b)² as the common factor]

= 8(a – b)² (2a – 2b – 3)

Question: 18

Factorize of the following expressions:

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) [taking (bx +a) as the common factor]

Question: 19

Factorize of the following expressions:

 a²x² + (ax² + 1)x + 1 + a

Solution:

a²x² + (ax² +1)x + 1 + a = a²x² + ax³ + x + a

= (ax³ + a²x²) + (x + a)

= ax²(x + a) + (x + a)

= (ax² + 1)(x + a) [taking (x + a) as the common factor]

Question: 20

Factorize of the following expressions:

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) [since, (c -a) = -(a – c)]

= a(a – c) – 2b(a -c)

= (a -2b)(a – c) [taking (a – c) as the common factor]

Question: 21

Factorize of the following expressions:

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) [taking (a – c) as the common factor]

Question: 22

Factorize of the following expressions:

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) [since, (1 – x) = – (x – 1)]

= (x – 11y)(x – 1) [taking out the common factor]

Question: 23

Factorize of the following expressions:

 ab – a – b + 1

Solution:

ab – a – b + 1 = (ab – b) + (1 – a)

= b(a – 1) + (1 – a)

= b(a – 1) – (a – 1) [since, (1-a) = -(a -1)]

= (a – 1)(b – 1) [taking out the common factor (a – 1)]

Question: 24

Factorize of the following expressions:

x² + y – xy – x

Solution:

x² + y – xy – x = (x² – xy) + (y -x)

= x(x – y) + (y – x)

= x(x – y) – (x – y) [(y – x) = -(x – y)]

= (x – 1)(x – y) [taking (x – y) as the common factor]