**Exercise 7.4**

**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) p**^{2}q – pr^{2} – pq + r^{2}

^{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)

(p – 1) as the common factor

= (pq – r^{2})(p – 1)

**(3) 1 + x + xy + x**^{2}y

^{2}y

**Solution:**

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

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

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

^{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})

(a^{2} + b^{2}) as the common factor

= (x – y)(a^{2} + b^{2})

**(6) x**^{2} + xy +xz + yz

^{2}+ xy +xz + yz

**Solution:**

x^{2} + xy +xz + yz

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

^{2}

**Solution:**

ab – by – ay + y^{2}

= (ab – ay) + (y^{2} – 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) lm**^{2} – mn^{2}– Lm + n^{2}

^{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)

Here (m – 1) a sthe common factor

= (lm – n^{2})(m – 1)

**(11) x**^{3} – y^{2} + x – x^{2}y^{2}

^{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)

Here (x^{2}+ 1) as the common factor

= (x – y^{2})(x^{2} + 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) x**^{2} – 2ax – 2ab + bx

^{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)

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

^{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)

**(15) abx**^{2} + (ay – b)x – y

^{2}+ (ay – b)x – y

**Solution:**

abx^{2} + (ay – b)x – y

= (abx^{2} – bx) + (axy – y)

= bx (ax – 1) + y(ax – 1)

= (bx + y)(ax – 1)

**(16) (ax + by)**^{2} + (bx – ay)^{2}

^{2}+ (bx – ay)

^{2}

**Solution:**

(ax + by)^{2} + (bx – ay)^{2}

(using identities: (a + b)^2 = a^{2} + b^{2} + 2ab ) and (a – b)^2 = a^{2} + b^{2} – 2ab ) )

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

**(17) 16(a – b)**^{3} – 24(a – b)^{2}

^{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(x**^{2} + 1) + x(a^{2} + b^{2})

^{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)

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

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

**(20) 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)

= a(a – c) – 2b(a -c)

= (a – 2b)(a – c)

**(21) 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)

**(22) x2 – 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)

= (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) x**^{2} + y – xy – 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)

= (x – 1)(x – y)