**Factorize each of the following algebraic expressions :**

**Q.1)Â 6x(2x – y) + 7y(2x – y)**

**Soln.:**

6x(2x – y) + 7y(2x – y)

= (6x + 7y)(2x – y)Â (taking (2x â€“ y) as common factor)

**Q.2)Â 2r(y – x) + s(x – y)Â **

**Soln.:**

2r(y – x) + s(x – y)

=Â 2r(y – x) â€“ s(y – x) [since, (x – y) = -(y – x)]

= (2r – s)(y – x)Â [taking (y – x) as the common factor]

**Q.3)Â 7a(2x – 3) + 3b(2x – 3) **

**Soln.:**

7a(2x – 3) + 3b(2x – 3)

= (7a + 3b)(2x – 3)Â [taking (2x – 3) as the common factor]

**Q.4)Â 9a(6a â€“ 5b) â€“ 12a ^{2}(6a â€“ 5b) **

**Soln.:**

9a(6a â€“ 5b) â€“ 12a^{2}(6a â€“ 5b)

= (9a â€“ 12qa^{2})(6a â€“ 5b)Â [taking (6a â€“ 5b) as the common factor]

= 3a(3 â€“ 4a)(6a â€“ 5b)Â [taking 3a as the common factor of the quadratic eqn. (9a â€“ 12a^{2})]

**Q.5)Â 5(x â€“ 2y) ^{2} + 3(x â€“ 2y)**

**Soln.:**

5(x â€“ 2y)^{2} + 3(x â€“ 2y)

= [(x â€“ 2y) + 3](x â€“ 2y) [taking (x â€“ 2y) as the common factor]

= (5x â€“ 10y + 3)(x â€“ 2y)

**Q.6)Â 16(2L â€“ 3m) ^{2} â€“ 12(3m â€“ 2L)**

**Soln.:**

16(2L â€“ 3m)^{2} â€“ 12(3m â€“ 2L)

= 16(2L â€“ 3m)^{2} + 12(2L â€“ 3m) [(3m â€“ 2L) =Â -(2L – 3m)]

= [16(2L â€“ 3m) + 12](2L â€“ 3m) [taking (2L – 3m) as the common factor]

= 4[4(2L â€“ 3m) + 3](2L â€“ 3m)Â [taking 4 as the common factor (16(2L â€“ 3m) + 12)]

= 4(8L â€“ 12m + 3)(2L â€“ 3m)

**Q.7)Â 3a(x â€“ 2y) â€“ b(x â€“ 2y)**

**Soln.:**

3a(x â€“ 2y) â€“ b(x â€“ 2y)

= (3aÂ -b)(x â€“ 2y)Â [taking (x â€“ 2y) as the common factor]

**Q.8)Â a ^{2}(x + y) + b^{2}(x + y) +c^{2}(x + y)**

**Soln.:**

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

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

**Q.9)Â (x – y) ^{2} + (x â€“ y)**

**Soln.:**

(x – y)^{2} + (x â€“ y)

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

= (x â€“ y + 1)(x – y)

**Q.10)Â 6(a + 2b) â€“ 4(a +2b) ^{2}**

**Soln.:**

6(a + 2b) â€“ 4(a +2b)^{2}

= [6 â€“ 4(a + 2b)](a + 2b) [taking (a + 2b as the common factor)]

= 2[3 â€“ 2(a + 2b)](a + 2b)Â [taking 2 as the common factor of [6 â€“ 4(a + 2b)]]

= 2(3 â€“ 2a â€“ 4b)(a + 2b)

**Q.11)Â a(x – y) + 2b(y – x) + c(x – y) ^{2}**

**Soln.:**

a(x – y) + 2b(y – x) + c(x – y)^{2}

= a(x â€“ y) â€“ 2b(x -y) +c(x – y)^{2} [(y -x) = -(x – y)]

= [a â€“ 2b + c(x- y)](x – y)

= (a â€“ 2b + cx – cy)(x- y)

**Q.12)Â -4(x – 2y) ^{2} + 8(x â€“ 2y)**

**Soln.:**

-4(x – 2y)^{2} + 8(x â€“ 2y)

= [-4(x â€“ 2y) + 8](x -2y)Â [taking (x â€“ 2y) as the common factor]

= 4[-(x â€“ 2y) + 2](x â€“ 2y) [taking 4 as the common factor of [-4(x â€“ 2y) + 8]]

= 4(2y â€“ x + 2)(x â€“ 2y)

**Q.13)Â x ^{3}(a â€“ 2b) + x^{2}(a â€“ 2b)**

**Soln.:**

x^{3}(a â€“ 2b) + x^{2}(a â€“ 2b)

= (x^{3} + x^{2})(a – 2b)Â [taking (a â€“ 2b) as the common factor]

= x^{2}(x + 1)(a â€“ 2b) [taking x^{2} as the common factor of (x^{3} + x^{2})]

**Q.14)Â (2x â€“ 3y)(a + b) + (3x – 2y)(a + b)**

**Soln.:**

(2x â€“ 3y)(a + b) + (3x – 2y)(a + b)

= (2x â€“ 3y + 3x â€“ 2y)(a +b)Â [taking (a +b) as the common factor]

= (5x â€“ 5y)(a + b)

= 5(x – y)(a + b)Â [taking 5 as the common factor of (5x â€“ 5y)]

**Q.15)Â 4(x + y)(3a – b) + 6(x + y)(2b â€“ 3a)**

**Soln.:**

4(x + y)(3a – b) + 6(x + y)(2b â€“ 3a)

= 2(x + y)[2(3a – b) + 3(2b â€“ 3a)] [taking (2(x + y)) as the common factor ]

= 2(x + y)(6a â€“ 2b + 6b â€“ 9a)

= 2(x + y)(4b â€“ 3a)