GCD OF ALGEBRAIC EXPRESSIONS

Example :

Find the greatest common divisor of following algebraic terms.

(i)  7 x² y z⁴, 21 x² y⁵ z³

(ii)  x² y, x³ y , x² y²

(iii)  25 b c⁴ d³ , 35 b² c⁵ , 45 c³ d

(iv)  35 x⁵ y³ z⁴ , 49 x² y z³ , 14 xy² z²

(i)  Answer :

7 x² y z⁴, 21 x² y⁵ z³

7 x² y z⁴   =  7 ⋅ x² ⋅ y ⋅ z ⋅ z³

21 x² y⁵ z³  =  3 ⋅ 7 ⋅ x² ⋅ y ⋅ z ⋅ z³

Common factors are 7, x²,  y and z³

Multiplying common factors, we get

=  7x²yz³

So, greatest common divisor of the given algebraic terms is  7x²yz^3.

(ii)  Answer :

x² y, x³ y , x² y²

x² y  =  x² ⋅ y

 x³ y  =  x² ⋅ x ⋅ y

 x² y²  =  x² ⋅ y ⋅ y

Common factors in the above terms are x² and y.

By multiplying common factors, we get x²y.

So, greatest common divisor of the given algebraic terms is x²y.

(iii)  Answer :

25 b c⁴ d³, 35 b² c⁵, 45 c³ d

25 b c⁴ d³  =  5 ⋅ 5 ⋅ b ⋅ c³ ⋅ d³

35 b² c⁵  =  7 ⋅ 5 ⋅ b ⋅ b ⋅ c³ ⋅ c²

45 c³ d  =  3² ⋅ 5 ⋅ c³ ⋅ d

Common factors in the above terms are 5 and c³.

By multiplying common factors, we get 5c³.

So, greatest common divisor of the given algebraic terms is  5c³.

(iv)  Answer :

35 x⁵ y³ z⁴ , 49 x² y z³ , 14 xy² z²

35 x⁵ y³ z⁴  =  7 ⋅ 5 ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ y ⋅ y ⋅ y ⋅ z² ⋅ z²

49 x² y z³  =   7 ⋅ 7 ⋅ x ⋅ x ⋅ y ⋅ z² ⋅ z

14 xy² z²  =   7 ⋅ 2 ⋅ x ⋅ y ⋅ y ⋅ z²

Common factors in the above terms are 7, x, y and z².

By multiplying common factors, we get 7xyz².

So, greatest common divisor of the given algebraic terms is 7xyz².

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