GCD OF ALGEBRAIC EXPRESSIONS
Greatest Common Factor (GCF) of two numbers is the greatest factor that is common to both of them
To find the greatest common divisor of the given numbers or for algebraic expressions we have to follow the steps.
Step 1 :
List the prime factors of each of the given number. For algebraic expression we have to find factors of them.
Step 2 :
List the common factors of the given numbers or common factors.
Step 3 :
Multiply those common factors.
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 7x2yz3.
(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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