Do you want to have a better understanding of factoring by grouping topics? Here, we are to help you in this and you have reached the correct place. By going through this article, you can clearly understand the factorization of the grouping method. We are offering the Worksheet on Factoring by Grouping, which includes different algebraic equations factorization with a simple solution. Practice using the factorization by grouping worksheets and enhance your conceptual knowledge.

We are providing all Factorization Worksheets for free of cost on our website. Check every worksheet and improve your preparation level.

## Solved Examples on Factoring by Grouping

1. Factor the following

(i) 12p + 15

(ii) 14p – 21

(iii) 9q – 12q²

## Solution:

(i) The given expression is 12p + 15.

Expand the given expression. That is,

(3 * 4)p + (3 * 5).

Factor out the greatest common factor. That is,

3(4p + 5).

Then, 12p + 15 is equal to 3(4p + 5).

(ii) The given expression is 14p – 21.

Expand the given expression. That is,

(7 * 2)p – (7 * 3).

Factor out the greatest common factor. That is,

7(2p – 3).

Then, 14p – 21 is equal to 7(2p – 3).

(iii) The given expression is 9q – 12q^2.

Expand the given expression. That is,

(3 * 3)q – (3 * 4)q^2.

Factor out the greatest common factor. That is,

3q(3 – 4q).

Then, 9q – 12q^2 is equal to 3q(3 – 4q).

2. Factor by grouping the expressions

(i) 16x² – 24xy

(ii) 15xy² – 20x²y

(iii) 12a²b³ – 21a³b²

## Solution:

(i) The given expression is 16x^2 – 24xy.

Expand the given expression. That is,

(4 * 4)x^2 – (4 * 6)xy.

Factor out the greatest common factor. That is,

4x(4x – 6y).

Then, 16x^2 – 24xy is equal to 4x(4x – 6y).

(ii) The given expression is 15xy^2 – 20x^2y.

Expand the given expression. That is,

(5 * 3)xy^2 – (5 * 4)x^2y.

Factor out the greatest common factor. That is,

5xy(3y – 4x).

Then, 15xy^2 – 20x^2y is equal to 5xy(3y – 4x).

(iii) The given expression is 12a^2b^3 – 21a^3b^2.

Expand the given expression. That is,

(3 * 4)a^2b^3 – (3 * 7)a^3b^2.

Factor out the greatest common factor. That is,

3a^2b^2(4b – 7a).

Then, 12a^2b^3 – 21a^3b^2 is equal to 3a^2b^2(4b – 7a).

3. Factorize the expressions

(i) 24a³ – 36a²b.

(ii) 10a³ – 15a²

(iii) 36a³b – 60a²b³c

## Solution:

(i) The given expression is 24a^3 – 36a^2b.

Expand the given expression. That is,

(6 * 4)a^3 – (6 * 6)a^2b.

Factor out the greatest common factor. That is,

6a^2(4a – 6b).

Then, 24a^3 – 36a^2b is equal to 6a^2(4a – 6b).

(ii) The given expression is 10a^3 – 15a^2.

Expand the given expression. That is,

(5 * 2)a^3 – (5 * 3)a^2.

Factor out the greatest common factor. That is,

5a^2(2a – 3).

Then, 10a^3 – 15a^2 is equal to 5a^2(2a – 3).

(iii) The given expression is 36a^3b – 60a^2b^3c.

Expand the given expression. That is,

(6 * 6)a^3b – (6 * 10)a^2b^3c.

Factor out the greatest common factor. That is,

6a^2b(6a – 10b^2c).

Then, 36a^3b – 60a^2b^3c is equal to 6a^2b(6a – 10b^2c).

4. Factorize

(i) 9a³ – 6a² + 12a.

(ii) 8a² – 72ab + 12a.

(iii) 18x³y³ – 27x²y³ + 36x³y²

## Solution:

(i) The given expression is 9a³ – 6a² + 12a.

Expand the given expression. That is,

(3 * 3)a^3 – (3 * 2)a^2 + (3 * 4)a.

Factor out the greatest common factor. That is,

3a(3a^2 – 2a + 4).

Then, 9a^3 – 6a^2 + 12a is equal to 3a(3a^2 – 2a + 4).

(ii) The given expression is 8a^2 – 72ab + 12a.

Expand the given expression. That is,

(4 * 2)a^2 – (4 * 18)ab + (4 * 3)a.

Factor out the greatest common factor. That is,

4a(2a – 18b + 3a).

Then, 8a^2 – 72ab + 12a is equal to 4a(2a – 18b + 3a).

(iii) The given expression is 18x³y³ – 27x²y³ + 36x³y²

Expand the given expression. That is,

(9 * 2)x^3y^3 – (9 * 3)x^2y^3 + (9 * 4)x^3y^2.

Factor out the greatest common factor. That is,

9x^2y^2(2xy -3y + 4x).

Then, 18x³y³ – 27x²y³ + 36x³y² is equal to 9x^2y^2 (2xy -3y + 4x).

5. How to factor by grouping?

(i) 14a³ + 21a^4b – 28a²b²

(ii) -5 – 10x + 20x²

## Solution:

(i) The given expression is 14a^3 + 21a^4b – 28a^2b^2.

Expand the given expression. That is,

(7 * 2)a^3 + (7 * 3)a^4b – (7 * 4)a^2b^2.

Factor out the greatest common factor. That is,

7a^2(2a + 3a^2b – 4b^2).

Then, 14a^3 + 21a^4b – 28a^2b^2 is equal to 7a^2(2a + 3a^2b – 4b^2).

(ii) The given expression is -5 – 10x + 20x^2.

Expand the given expression. That is,

-5 – (5 * 2)x + (5 * 4)x^2.

Factor out the greatest common factor. That is,

5( -1 – 2x + 4x^2).

Then, -5 – 10x + 20x^2 is equal to 5( -1 – 2x + 4x^2) .

6. Factoring

(i) a (a + 3) + 5(a + 3)

(ii) 5a (a – 4) – 7 (a – 4).

(iii) 2x (1 – y) + 3(1 – y).

## Solution:

(i) The given expression is a(a + 3) + 5(a + 3).

Factor out the greatest common factor. That is,

(a + 3) (a + 5)

Then,a (a + 3) + 5(a + 3) is equal to (a + 3) (a + 5).

(ii) The given expression is 5a(a – 4) – 7(a – 4).

Factor out the greatest common factor. That is,

(a – 4) (5a – 7).

Then, 5a (a – 4) – 7 (a – 4) is equal to (a – 4) (5a – 7).

(iii) The given expression is 2x(1 – y) + 3(1 – y).

Factor out the greatest common factor. That is,

(1 – y) (2x + 3).

Then, 2x(1 – y) + 3(1- y) is equal to (1 – y) (2x + 3).

7. Factoring the expressions

(i) 6x (x – 2y) + 5y (x – 2y).

(ii) p³ (2x – y) + p²(2x – y).

## Solution:

(i) The given expression is 6x (x – 2y) + 5y (x – 2y).

Factor out the greatest common factor. That is,

(x – 2y) (6x + 5y).

Then, 6x (x – 2y) + 5y (x – 2y) is equal to (x – 2y) (6x + 5y).

(ii) The given expression is p^3 (2x – y) + p^2 (2x – y).

Factor out the greatest common factor. That is,

(2x – y)p^2 (p + 1).

Then, p^3 (2x – y) + p^2 (2x – y) is equal to p^2(2x – y) (p + 1).

8. How to factor by grouping polynomials?

(i) 9x (3x – 5y) – 12x²(3x – 5y)

(ii) (a + 5)² – 4 (a + 5)

(iii) 3(x – 2y)² – 5(x – 2y)

## Solution:

(i) The given expression is 9x (3x – 5y) – 12x²(3x – 5y).

Factor out the greatest common factor. That is,

3x(3x – 5y)(3 – 4x).

Then, 9x (3x – 5y) – 12x²(3x – 5y) is equal to 3x(3x – 5y)(3 – 4x).

(ii) The given expression is (a + 5)^2 – 4 (a + 5).

Factor out the greatest common factor. That is,

(a + 5) [(a + 5) – 4] = (a + 5) (a + 1).

Then, (a + 5)^2 – 4 (a + 5) is equal to (a + 5) (a + 1).

(iii) The given expression is 3(x – 2y)² – 5(x – 2y).

Factor out the greatest common factor. That is,

(x – 2y) [3(x – 2y) – 5].

Then, 3(x – 2y)² – 5(x – 2y) is equal to (x – 2y) [3(x – 2y) – 5].

9. Factor completely

(i) 2x + 6y – 3(x + 3y)²

(ii) 16(2x – 3y)² – 4(2x – 3y)

(iii) p (x – 3) + q (3 – x)

(iv) 12 (2a – 3b)² – 16 (3b – 2a).

(v) (a + b)(2a + 5) – (a + b)(a + 3).

## Solution:

(i) The given expression is 2x + 6y – 3(x + 3y)^2.

Factor out the greatest common factor. That is,

Factor out the greatest common factor. That is,

2(x + 3y) – 3(x + 3y)^2 = (x + 3y) (2 – 3(x + 3y)).

Then, 2x + 6y – 3(x + 3y)^2 is equal to (x + 3y) (2 – 3(x + 3y)).

(ii) The given expression is 16(2x – 3y)² – 4(2x – 3y).

Factor out the greatest common factor. That is,

4(2x – 3y) [4(2x – 3y) – 1].

Then, 16(2x – 3y)² – 4(2x – 3y) is equal to 4(2x – 3y) [4(2x – 3y) – 1].

(iii) The given expression is

p (x – 3) + q (3 – x).

we can write it as p (x – 3) – q(x – 3).

Factor out the greatest common factor. That is,

(x – 3) (p – q).

Then, p (x – 3) + q (3 – x) is equal to (x – 3) (p – q).

(iv) The given expression is 12 (2a – 3b)² – 16 (3b – 2a).

Expand the above expression. That is,

12 (2a – 3b)^2 + 32a – 48b.

We can write it as 12 (2a – 3b)^2 + 16 (2a – 3b).

Factor out the greatest common factor. That is,

4(2a – 3b) [3(2a – 3b) + 4].

Then, 12 (2a – 3b)² – 16 (3b – 2a) is equal to 4(2a – 3b) [3(2a – 3b) + 4].

(v) The given expression is (a + b)(2a + 5) – (a + b)(a + 3).

Factor out the greatest common factor. That is,

(a + b) [(2a + 5) – (a + 3)] = (a + b) (a + 2).

Then, (a + b)(2a + 5) – (a + b)(a + 3) is equal to (a + b) (a + 2).

10. How to factor by grouping polynomials?

(i) xp + yp + xq + yq.

(ii) p² – sp – qp + sq.

(iii) xy² – yz² – xy + z²

(iv) a² – ac + ab–bc.

(v) 6xy – y² + 12xz – 2yz.

## Solution:

(i) The given expression is xp + yp + xq + yq.

Factor out the greatest common factor. That is,

p(x + y) + q(x + y) = (x + y) (p + q).

Then, xp + yp + xq + yq is equal to (x + y) (p + q).

(ii) The given expression is p2 – sp – qp + sq.

Factor out the greatest common factor. That is,

p(p – s) – q(p – s) = (p – s) (p – q).

Then, p2 – sp – qp + sq is equal to (p – s) (p – q).

(iii) The given expression is xy² – yz² – xy + z²

Factor out the greatest common factor. That is,

xy^2 – xy – yz^2 + z^2 = xy(y – 1) – z(yz – z).

Then, xy² – yz² – xy + z² is equal to xy(y – 1) – z(yz – z).

(iv) : The given expression is a2 – ac + ab – bc.

Factor out the greatest common factor. That is,

a(a – c) + b(a – c) = (a – c) (a + b).

Then, a2 – ac + ab – bc is equal to (a – c) (a + b).

(v) The given expression is 6xy – y² + 12xz – 2yz.

Factor out the greatest common factor. That is,

y(6x – y) + 2z(6x – y) = (6x – y) (y + 2z).

Then, 6xy – y² + 12xz – 2yz is equal to (6x – y) (y + 2z).

11. Solve by factoring

(i) (a – 2b)² + 4a–8b

(ii) b² – ab (1 – a) – a³

(iii) (rp + sq)² + (sp – rq)²

(iv) pq² + (p – 1)q – 1

## Solution:

(i) The given expression is (a – 2b)² + 4a–8b.

Factor out the greatest common factor. That is,

(a – 2b)² + 4(a – 2b) = (a – 2b) (a – 2b + 4).

Then, (a – 2b)² + 4a–8b is equal to (a – 2b) (a – 2b + 4).

(ii) The given expression is b² – ab (1 – a) – a³

we can write it as b^2 – ab + a^2b – a^3.

Factor out the greatest common factor. That is,

b(b – a) + a^2(b – a) = (b – a)(b + a^2).

Then, b² – ab (1 – a) – a³ is equal to (b – a)(b + a^2).

(iii) The given expression is (rp + sq)² + (sp – rq)²

We can write it as (rp)^2 + (sq)^2 + 2rpsq + (sp)^2 + (rq)^2 – 2rpsq

it is equal to (rp)^2 + (sq)^2 + (sp)^2 + (rq)^2

Factor out the greatest common factor. That is,

r^2(p^2 + q^2) + s^2(p^2 + q^2) = (p^2 + q^2) (r^2 + s^2)

Then, (rp + sq)² + (sp – rq)² is equal to (p^2 + q^2) (r^2 + s^2)

(iv) The given expression is pq² + (p – 1)q – 1.

we can write it as pq^2 + pq – q – 1= pq(q +1) – (q + 1)

Factor out the greatest common factor. That is,

(q + 1) (pq – 1).

Then, pq² + (p – 1)q – 1 is equal to (q + 1) (pq – 1).

12. Factoring Algebraic Expressions

(i) p³ – 3p² + p – 3

(ii) xy (p² + q²) – pq (x² + y²)

(iii) p² – p(x + 2y) + 2xy

## Solution:

(i) The given expression is p^{3} – 3p^{2} + p – 3.

We can write it as p^2(p – 3) + (p – 3)

Factor out the greatest common factor. That is, (p – 3) (p^2 + 1).

Then, p^{3} – 3p^{2} + p – 3is equal to (p – 3) (p^2 + 1).

(ii) The given expression is xy (p^{2} + q^{2}) – pq (x^{2} + y^{2}).

We can write it as xyp^2 + xyq^2 – pqx^2 – pqy^2.

Factor out the greatest common factor. That is, xp (yp –qx) – yq (yp–qx) = (yp – qx) (xp – yq).

Then, xy (p^{2} + q^{2}) – pq (x^{2} + y^{2})is equal to (yp – qx) (xp – yq).

(iii) The given expression is p^{2} – p(x + 2y) + 2xy.

We can write it as p^2 – px – 2py + 2xy.

Factor out the greatest common factor. That is, p(p -2y) – x(p – 2y) = (p – 2y) (p – x).

Then, p^{2} – p(x + 2y) + 2xy is equal to (p – 2y) (p – x).