GREATEST COMMON FACTOR OF POLYNOMIALS WORKSHEET

Problem 1 : 

Find the greatest common factor of the following : 

14xy²   and   42xy

Problem 2 : 

Find the greatest common factor of the following : 

16x³y²   and   24xy³z

Problem 3 : 

Find the greatest common factor of the following : 

(y³ + 1)   and   (y³ - 1)

Problem 4 : 

Find the greatest common factor of the following : 

(a³ - 27)   and   (a² - 6a + 9)

Problem 5 : 

Find the greatest common factor of the following : 

(2x² - 18)   and   (x² - 2x - 3)

Problem 6 : 

Find the greatest common factor of the following : 

(a - b)², (b - c)³, (c - a)⁴

Detailed Answer Key

Problem 1 : 

Find the greatest common factor of the following : 

14xy²   and   42xy

Solution : 

Factor :

14xy²  =  2 ⋅ 7 ⋅ x ⋅ y ⋅ y

42xy  =  2 ⋅ 3 ⋅ 7 ⋅ x ⋅ y

Therefore, the greatest common factor of 14xy² and 42xy is

=  2 ⋅ 7 ⋅ x ⋅ y

=  14xy

Problem 2 : 

Find the greatest common factor of the following : 

16x³y²   and   24xy³z

Solution :

16x³y²  =  2⁴ ⋅ x³ ⋅ y²  =   2³ ⋅ 2 ⋅ x² ⋅ x ⋅ y²

  24xy³z  =  2³ ⋅ 3 ⋅ x ⋅ y³ ⋅ z  =  2³ ⋅ 3 ⋅ x ⋅ y² ⋅ y ⋅ z  

So, the greatest common factor is 

=  2³ ⋅ x ⋅ y²

=  8xy²

Problem 3 : 

Find the greatest common factor of the following : 

(y³ + 1)   and   (y³ - 1)

Solution :

Algebraic Identities : 

a³ + b³  =  (a + b)(a² - ab + b²)

a² - b²  =  (a + b)(a - b)

Using the above identities, factor the given polynomials and find the greatest common factor. 

y³ + 1  =  y³ + 1³  =  (y + 1)(y² - y + 1)

y² - 1  =  y² - 1²  =  (y + 1)(y - 1)

So, the greatest common factor is 

=  (y + 1)

Problem 4 : 

Find the greatest common factor of the following : 

(a³ - 27)   and   (a² - 6a + 9)

Solution :

Factor the given polynomials and find the greatest common factor. 

Factor (a³ - 27) : 

a³ - 27  =  a³ - 3³

Use algebraic identity a³ - b³  =  (a - b)(a² - ab + b²).

a³ - 27  =  (a - 3)(a² - 3a + 3²)

a³ - 27  =  (a - 3)(a² - 3a + 9) -----(1)

Factor (a² - 6a + 9) : 

a² - 6a + 9  =  a² - 3a - 3a + 9 

a² - 6a + 9  =  a(a - 3) - 3(a - 3)

a² - 6a + 9  =  (a - 3)(a - 3) -----(2) 

From (1) and (2), the greatest common factor is 

(a - 3)

Problem 5 : 

Find the greatest common factor of the following : 

(2x² - 18)   and   (x² - 2x - 3)

Solution :

Factor the given polynomials and find the greatest common factor. 

Factor (2x² - 18) : 

2x² - 18  =  2(x² - 9)

2x² - 18  =  2(x² - 3²)

2x² - 18  =  2(x + 3)(x - 3) -----(1)

Factor (x² - 2x - 3) : 

x² - 2x - 3  =  x² - 3x + x - 3

x² - 2x - 3  =  x(x - 3) + 1(x - 3)

x² - 2x - 3  =  (x - 3)(x + 1) -----(2)

From (1) and (2), the greatest common factor is 

(x - 3)

Problem 6 : 

Find the greatest common factor of the following : 

(a - b)², (b - c)³, (c - a)⁴

Solution :

For the given polynomials, there is no common factor other than one. 

So, the greatest common factor is 1. 

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