HIGHEST COMMON FACTOR OF ALGEBRAIC EXPRESSIONS

Find the HCF of :

Problem 1 :

5(a + 7) and (a + 1) (a + 7)

Solution :

Factor of 5(a + 7) and (a + 1) (a + 7)

5(a + 7) = 5 ⋅ (a + 7)

(a + 1) (a + 7) = (a + 1) ⋅ (a + 7)

= (a + 7)

HCF of 5(a + 7) and (a + 1) (a + 7) is (a + 7).

Problem 2 :

3(2 + b)² and 3(6 + b) (2 + b)

Solution :

Factor of 3(2 + b)² and 3(6 + b) (2 + b)

3(2 + b)² = 3 ⋅ (2 + b) ⋅ (2 + b)  

3(6 + b) (2 + b) = 3 ⋅ (6 + b) ⋅ (2 + b) 

= 3 ⋅ (2 + b) 

HCF of 3(2 + b)² and 3(6 + b) (2 + b) is 3(2 + b).

Problem 3 :

x²(x – 4) and x(x – 4)

Solution :

Factor of x²(x – 4) and x(x – 4)

x²(x – 4) = x ⋅ x ⋅ (x – 4)

 x(x – 4) = x ⋅ (x – 4)

= x ⋅ (x – 4)

HCF of x²(x – 4) and x(x – 4) is x(x – 4).

Problem 4 :

15(x – 2)² and 5(x – 2) (x + 3)

Solution :

Factor of 15(x – 2)² and 5(x – 2) (x + 3)

15(x – 2)² = 5 ⋅ 3 ⋅ (x – 2) ⋅ (x – 2)

5(x – 2) (x + 3) = 5 ⋅ (x – 2) ⋅ (x + 3)

= 5 ⋅ (x – 2)

HCF of 15(x – 2)² and 5(x – 2) (x + 3) is 5(x – 2).

Problem 5 :

6(x – 1)² and 16(x – 7) (x - 1)

Solution :

Factor of 6(x – 1)² and 16(x – 7) (x - 1)

6(x – 1)² = 3 ⋅ 2 ⋅ (x – 1) ⋅ (x – 1)

 16(x – 7) (x - 1) = 8 ⋅ 2 ⋅ (x – 7) ⋅ (x – 1)

= 2 ⋅ (x – 1)

HCF of 6(x – 1)² and 16(x – 7)(x - 1) is 2(x – 1).

Problem 6 :

9y(y + 1) and 6y(y + 1)²

Solution :

Factor of 9y(y + 1) and 6y(y + 1)²

9y(y + 1) = 3 ⋅ 3 ⋅ y ⋅ (y + 1)

6y(y + 1)² = 2 ⋅ 3 ⋅ y ⋅ (y + 1) ⋅ (y + 1)

= 3 ⋅ y ⋅ (y + 1)

HCF of 9y(y + 1) and 6y(y + 1)² is 3y(y + 1).

Problem 7 :

Find the Greatest Common Factor for the given expressions

a) 10p³ q, 12 pq²

b) 8a² b³, 10ab²

c) 12m² n³, 30 m⁵ n³

d) 28x² y⁴, 42 x⁴ y⁴

e) 10a³, 12 a², 14a

Solution :

a) 

10p³ q = 2 ⋅ 5 p³ q

12 pq² = 2² ⋅ 3 ⋅ p ⋅ q³

Greatest common factor =  2 ⋅ p ⋅ q

= 2pq

b) 8a² b³, 10ab²

8a² b³ = 2 ⋅ 2 ⋅ 2 a² ⋅ b³ 

10ab² = 2 ⋅ 5 ⋅ a ⋅ b²

Greatest common factor =  2 ⋅ a ⋅ b²

= 2a b²

c) 

12m² n³ = 2² ⋅ 3 m² n³

30 m⁵ n³ = 2 ⋅ 3 ⋅ 5 m⁵ n³

Greatest common factor = 2 ⋅ 3 m² n³

= 6 m² n³

d)

28x² y⁴ = 2² ⋅ 7 x² y⁴

42 x⁴ y⁴ = 2 ⋅ 3 ⋅ 7 x⁴ y⁴

Greatest common factor = 2⋅ 7 x² y⁴

= 14 x² y⁴

e)

10a³ = 2 ⋅ 5a³

12 a² = 2² ⋅ 3a²

14a = 2 ⋅ 7a

Greatest common factor = 2a

Problem 8 :

In the following exercises, factor the greatest common factor from each polynomial.

a) 6m + 9

b) 9n - 63

c) 24x³ - 12x² + 15x

Solution :

a)

6m + 9 = 2 ⋅ 3 ⋅ m + 3 ⋅ 3

Greatest common factor for these two terms is 3, factoring it out

= 3(2m + 3)

b)

9n - 63 = 9 ⋅ n - 9 ⋅ 7

Greatest common factor is 9, then factoring it out

= 9(n - 7)

c)

24x³ - 12x² + 15x 

Here considering the coefficients, these are multiples of 3. Factoring 3x, we get

= 3x(8x² - 4x + 5)

Problem 9 :

The area (in square centimeters) of a square coaster can be represented by

d² + 8d + 16

a. Write an expression that represents the side length of the coaster.

b. Write an expression for the perimeter of the coaster.

Solution :

= d² + 8d + 16

= d² + 4d + 4d + 16

= d(d + 4) + 4(d + 4)

= (d + 4)(d + 4)

a) side length of the coaster is d + 4 units

b) Perimeter = 4(d + 4)

= 4d + 16

Problem 10 :

The polynomial represents the area (in square feet) of the square playground.

a. Write a polynomial that represents the side length of the playground.

b. Write an expression for the perimeter of the playground.

Solution :

= x² - 30x + 225

= x² - 15x - 15x + 225

= x(x - 15) - 15(x - 15)

= (x - 15)(x - 15)

a) Side length of the play ground is x - 15 units

b) Perimeter of the play ground = 4(x - 15)

= 4x - 60 units.