**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:

p^{2}q – pr^{2} – pq + r^{2}

**Solution:**

p^{2}q – pr^{2} – pq + r^{2}

= (p^{2}q – pq) + (r^{2} – pr^{2})

pq(p – 1) + r^{2}(1 – p)

pq(p – 1) – r^{2}(p – 1) [since, (1 – p) = -(p – 1)]

= (pq – r^{2})(p – 1) [taking (p – 1) as the common factor]

**Question: 3**

Factorize of the following expressions:

1 + x + xy + x^{2}y

**Solution:**

1 + x + xy + x^{2}y

= (1 + x) + (xy + x^{2}y)

= (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:

xa^{2 }+ xb^{2} – ya^{2} – yb^{2}

**Solution:**

xa^{2 }+ xb^{2} – ya^{2} – yb^{2}

= (xa^{2} + xb^{2}) – (ya^{2} + yb^{2})

= x(a^{2} + b^{2}) – y(a^{2} + b^{2})

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

**Question: 6**

Factorize of the following expressions:

x^{2} + xy + xz + yz

**Solution:**

x^{2} + xy + xz + yz

= (x^{2} + 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 + y^{2}

**Solution:**

ab – by – ay + y^{2}

= (ab – ay) + (y^{2} – 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:

Lm^{2} – mn^{2}– Lm + n^{2}

**Solution:**

Lm^{2} – mn^{2}– Lm + n^{2} = (Lm^{2} – Lm) + (n^{2} – mn^{2})

= Lm(m – 1) + n^{2}(1 – m)

= Lm(m – 1) – n^{2}(m – 1) [since, (1 – m) = -(m – 1)]

= (Lm – n^{2})(m – 1) [taking (m – 1) as the common factor]

**Question: 11**

Factorize of the following expressions:

x^{3 }– y^{2} + x – x^{2}y^{2}

**Solution:**

x^{3 }- y^{2} + x – x^{2}y^{2}

= (x^{3} + x) – (x^{2}y^{2} + y^{2})

= x(x^{2} + 1) – y^{2}(x^{2} + 1)

= (x – y^{2})(x^{2 }+ 1) [taking (x^{2 }+ 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:

x^{2} – 2ax – 2ab + bx

**Solution:**

x^{2} – 2ax – 2ab + bx

= (x^{2} – 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:

x^{3 }– 2x^{2}y + 3xy^{2} – 6y^{3}

**Solution:**

x^{3 }– 2x^{2}y + 3xy^{2} – 6y^{3}

= (x^{3} – 2x^{2}y) + (3xy^{2} – 6y^{3})

= x^{2}(x – 2y) + 3y^{2}(x – 2y)

= (x^{2} + 3y^{2})(x – 2y) [taking (x – 2y) as the common factor]

**Question: 15**

Factorize of the following expressions:

abx^{2 }+ (ay – b)x – y

**Solution:**

abx^{2 }+ (ay – b)x – y = abx^{2} + axy – bx – y

= (abx^{2} – 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} = a^{2}x^{2} + 2abxy + b^{2}y^{2 }+ b^{2}x^{2} – 2abxy + a^{2}y^{2}

= a^{2}x^{2} + b^{2}y^{2} + b^{2}x^{2} + a^{2}y^{2}

= (a^{2}x^{2 }+ a^{2}y^{2}) + (b^{2}x^{2} + b^{2}y^{2})

= a^{2}(x^{2 }+ y^{2}) + b^{2’}(x^{2 }+ y^{2})

= (a^{2 }+ b^{2})(x^{2 }+ y^{2}) [taking (x^{2} + y^{2}) 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(x^{2} + 1) + x(a^{2} + b^{2})

**Solution:**

ab(x^{2} + 1) + x(a^{2} + b^{2}) = abx^{2} + ab + a^{2}x + b^{2}x

= (abx^{2} + a^{2}x) + (b^{2}x + 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:

a^{2}x^{2} + (ax^{2} + 1)x + 1 + a

**Solution:**

a^{2}x^{2} + (ax^{2} +1)x + 1 + a = a^{2}x^{2} + ax^{3} + x + a

= (ax^{3} + a^{2}x^{2}) + (x + a)

= ax^{2}(x + a) + (x + a)

= (ax^{2} + 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 = a^{2} – 2ab – ac + 2bc

= (a^{2} – 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 = a^{2} + ab – ac – bc

= (a^{2} – 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:

x^{2 }- 11xy - x + 11y

**Solution:**

x^{2} – 11xy -x + 11y = (x^{2} – 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:

x^{2} + y – xy – x

**Solution:**

x^{2} + y – xy – x = (x^{2} – 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]